The word being translated as 'rite' is in fact karman, which is literally 'action, work'. However here it signifies a ritual action, hence we translate it as 'rite'. This is the first of several clear links with the Vedic sacrificial ritual.
The four rites in Sanskrit  are:
|śāntikakarman||rite of pacification|
|vaśyakarman||rite of subjection|
|puṣṭikarman||rite of prospering|
|raudrakarman||fierce rite, rite of destruction|
In early tantric texts such as Mahāvairocana Abhisaṃbodhi Tantra the various rites are actually forms of 'homa' (Chinese/Japanese goma) ritual. The word 'homa' means 'the act of making an oblation' and dervies from the root √hu 'sacrifice'. In Vedic ritual this function was carried out by the hotṛ priest . The Buddhist homa ritual involves setting up a sacrificial alter with a fire, and making coloured offerings to the fire. In Vedic times the idea was that the similarity between the microcosm and macrocosm allowed one to be influenced by the other through the ritual (which occupied a kind of intermediate space). In particular fire (agni) transformed the offerings into smoke which then wafted to heaven and induced the deva to respond (this kind of connected thinking underpins tantric sādhana as well). In the homa ritual the correspondence is between the body, speech and mind of the devotee and the Three Mysteries (triguhya) of the Dharmakāya Buddha which also have body, speech and mind aspects: all forms are the body, all sounds the speech, and all mental activity the mind of the Dharmakāya. The ritual conceives of the fire altar as an analogue for both (the altar itself is the body, the hearth is speech, and the fire is the mind), and through the ritual the microcosm of the individual is brought into with the macrocosm of the Dharmakāya. This kind of imagery is also drawing on Vedic models, but Buddhists are always careful to insist that śūnyatā (lack of self-nature) and pratītya-samutpāda (dependent-arising) underpin all their practices - so one is not merging with God, or with a numinous universal principle, but directly realising śūnyatā.
For the early Vedic priests the desired response of the ritual was keeping the natural order by bringing the rains at the proper time and averting disasters, but it was also connected with the health and prosperity of the king. In the tantric rites it is the individual who benefits and if there is a spiritual purpose to them, then it appears to be grafted onto the mundane, rather than the other way around. That is to say that it appears to me that these rites were already being used for mundane purposes when Buddhists began to adapt them for spiritual purposes, and that the mundane, even vulgar, use has been retained. We find mention of some of the rites in Gṛhyasūtras which covered domestic rites in Brahmin households. 
Each of the rites is associated with a colour and here too the rites tell us of their Vedic origins because the colours are: white, red, yellow and black. These are the colours associated in the Vedic tradition with the four varṇa or classes.  In fact varṇa more literally means 'colour'. So the brahmaṇa was associated with white symbolising their purity and the śudra with black symbolising their impurity (as I mentioned in A Pāli Pun). The kṣatriya were symbolised by red, and vaiṣya by yellow. The functions of the rites relate to some extent to the classes as well. Brahmins were concerned with rites and rituals, and ritual purity; kṣatriyas with ruling and conquering; vaiṣyas with agriculture and commerce; and śudras were serfs forced to labour. So we get these correspondences:
Why śudra and destruction? It may be that the impurity of the śudras threatened the makers of the original system; or that the were perceived as barbarous. Rudra, from which raudra ("connected with Rudra", "destruction") derives, is the name of a Vedic god who by this time was associated with Śiva who is also known as 'the destroyer' because his role in the Hindu trinity of gods is to preside over the destruction phase at the end of each time cosmic epoch (Brahmā is the creator, and Viṣṇu the sustainer). Perhaps some of the śudras worshipped Śiva?
These rites were absorbed into the Buddhist tradition at the time of the great synthesis and renewal which we call 'Tantra'.  Since they appear in the Mahāvairocana Abhisaṃbodhi Tantra we know that they must have been incorporated near the beginning of the process since this text is the earliest systematic tantra, and is thought to have been composed sometime in the 640's CE. In this text each of the rites consists of a pūjā which involves a series of preparatory practices in which one visualises oneself as a Buddha, the creation of maṇḍala with a fire place in the middle, an invocation to the deva Agni, and then the offering of appropriated coloured offerings accompanied by mantras. Some time much later the various functions were incorporated into mantras of White Tārā and I have written about some of these on my other website: visiblemantra.org - White Tārā. See especially the section: Other forms of the mantra.
Such rituals are still regularly carried out by both Tibetan and Japanese Vajrayāna practitioners, as well as some Hindu devotees. The goals of such rituals vary. I think on the whole they are used for spiritual purposes in the present day. But Stephen Beyer notes some mundane uses of such rites: So for instance he records:
"...and within my experience [Kurukullā's rite of subjugation] has been called upon by at least one Tibetan refugee group to coerce the Indian government. Tibetan traders seeking profit and Tibetan lovers seeking satisfaction followed upon the the ritual tracks established by their Indian processors." [The Cult of Tara: Magic and Ritual in Tibet, p. 302]
It may be this kind of behaviour which lead David Snellgrove to comment:
"So far as the verbal expression is concerned the most suitable English word for all these Sanskrit [synonyms for mantra] is undoubtedly 'spell.' One attracts by a spell, one binds by a spell, one releases by a spell... whether one likes it or not, the greater part of the tantras were concerned precisely with vulgar magic..." [Indo-Tibetan Buddhism, p.143]So these rites began as popular adaptations of the larger and more complex Vedic fire rituals, and from there were adapted by Buddhists, and to some extent they retained their 'vulgar' purposes. Martin Willson's introductory notes on the Tārā Tantra suggest caution with respect to the rites as found in the texts:
But someone has been playing a practical joke on Tibetan would-be magicians for the last eight centuries - the mantras have been shuffled. Anyone who thought he was summoning a woman with the rite of Chapter 16 was actually driving her away... At best the arrangement of the other mantras is uncertain. [In Praise of Tara: Songs to the Saviouress, p.48]As I say there is a living tradition, dating back to the mid 7th century, of performing these rites in a bona fide spiritual context in both Tibetan and Japanese vajrayāna circles - and while the Tārā Tantra may be muddled the Mahāvairocana Abhisaṃbodhi Tantra does not appear to be. Magic they may be, but the vulgar tag doesn't apply generally, and Snellgrove appears to have overstated his case.
Later the rituals were adapted to better fit the Five Buddha Maṇḍala with colours matching them: white, blue, yellow, red, and green with a surrounding aura of a special colour known in Tibet as chenka (it is said to be indescribable, but something like amethyst).  The function of the rites are then modified to better fit the functions of the five Buddhas. Subjugation for instance, becomes more like 'fascination' to fit with Amitābha's pratyavekṣana-jñāna or wisdom of discrimination, and his compassion. The rite of destruction is no longer for killing people, but for overcoming hindrances to practice and so on. It was this more wholesome set that I wrote about in my original essays on the rites.
The tantric rites are a good example of the eclecticism of tantra and of Indian religion generally. I've commented on this before, but it is worth saying again that in the Indian context this is far from unusual: in fact it is the norm. It is only from the point of view of strict monotheism that such borrowings look odd. This is not quite the same thing as saying all religions are the same, or that one can put together any religious elements and have a viable spiritual path. However it does mean that practices from another faith might be employed in Buddhism, although there is usually a thorough re-contextualisation of any new material, and at the same time religion (including monotheism) can be and often is subverted for mundane and vulgar purposes.
Sangharakshita has presented tantric material to the Triratna movement in terms of it's symbolism, for instance in his book Creative Symbols of Tantric Buddhism (which discusses the four rites in the section on colour symbolism), without directly passing on tantric teachings he received from his Tibetan teachers. Although we make use of tantric symbolism - somewhat naïvely I would argue - we are not a tantric movement. A few members of our Order who take tantra more seriously - notably Vessantara and Prakaśa - have sought abhiṣekha with Tibetan teachers. On the other hand Sangharakshita has written polemically about the breakdown of the proper guru/disciple relationship in Tibetan Buddhism and is scathing about people who collect initiations, and teachers who give them to anyone who asks or is willing to pay the fee. (Whereas I would argue that the function of giving of initiations has naturally shifted in the displaced Tibetan community and that this hardly represents a degradation but is a cultural adaptation to very difficult circumstances, and is in any case less radical than Sangharakshita's own de-contextualisation of tantra.)
My earlier essays on the rites: white, blue, yellow, red, green.
Another good source of info relevant to the Triratna Order's approach to the Tantric rites is Subhuti's talks on Kalyana Mitrata, published by Padmaloka and still on sale for £4.50. Unfortunately when these talks were republished as Buddhism and Friendship, the Tantric Rites sections were omitted.
- The Tibetan equivalents are: śāntikarma: zhi-ba’i ‘phrin-las; puṣṭikarma: rgyas-pa’i phrin-las; vaiṣyakarma: dbang gi phrin-las; raudrakarma: drag-po’i phrin-las.
- The hotṛ was one of four types of priest: three each associated with the three vedas, and a fourth, the brāhmaṇa, who was an overseer and put right any errors. The word hotṛ is the root hu with the -tṛ suffix making it an agent noun, and so means 'the sacrificer'.
- The Gobhila-Gṛhyasūtra for instance mentions the puṣṭikarma. It is also found in the Kausikapaddhati which is an 11th century commentary on the Atharvaveda. The śāntikarma is mentioned in the Āśvalāyana-Gr̥hyasūtra. There are several mentions in the Mahābharata.
- I use class to translation varṇa even though many scholars use caste. This is because caste more properly relates to jāti (the word is the same in historic Sanskrit, and in present day North Indian languages). While there are only four varṇa, there are now thousands of jāti. The division of society in terms of jāti was well in place by the time tantra began to develop. Indeed later tantra specifically negates Brahmanical class purity boundaries by contact with and ingesting of ritual impure substances.
- In fact some of the rituals described in the Suvarṇabhāsottama (Golden Light) Sūtra resemble Hindu rituals to some extent as well, which indicates that some intermixing may have occurred earlier without necessarily implying that Tantric Buddhism predates the 7th century.
- I recall reading about this somewhere but now that I come to reference it, I cannot find a single source. So either I made it up, or my recollection of the spelling is hopelessly out.
Image: a Shingon monk performing the homa ritual.