Veṅkaṭanātha (traditional dates 1269–1370 (see Neevel 1977 for a convincing explanation of these too long life spans) is a complex figure who can be interpreted in different ways according to the facet one is focusing on. What is sure is that what we refer to as Viśiṣṭādvaita Vedānta has been largely influenced by the shape he gave to it. For instance, the traditional lineage of teachers of Viśiṣṭādvaita Vedānta groups together teachers who bear some vague family resemblance among each other, but who are all directly linkable to Veṅkaṭanātha’s view of Viśiṣṭādvaita Vedānta.
The main philosophical outlines of Veṅkaṭanātha’s Viśiṣṭādvaita Vedānta are:

1. the Vedāntic viewpoint
2. the emphasis on Pūrva Mīmāṃsā
3. the incorporation of Pāñcarātra
4. the incorporation of the Āḻvārs’ theology

The development of Viśiṣṭādvaita Vedānta as a vedāntic school is clear as one looks back at Veṅkaṭanātha’s predecessors, but it is important to notice that what seems a posteriori like a clear Vedāntic school would not probably have appeared as such to its contemporaries. In fact, Yāmuna’s relation to Vedānta is complex. He quotes from the Upaniṣads in the Ātmasiddhi, and he starts it listing the Vedānta teachers he wants to refute (including Bhartṛhari and Śaṅkara), so that one might think that he is keener to “purify” Vedānta than he cares about “purifying” Nyāya. At the same time, at least in the Ātmasiddhi (see Mesquita 1971, pp. 4–13) Yāmuna accepts an anti-Vedāntic proof for the existence of God, namely the inference (whereas Pūrva and Uttara Mīmāṃsā agree that God can only be known through the Sacred Texts) and the Āgamaprāmāṇya seems to have a completely different focus. Rāmānuja is more straightforwardly part of a Vedāntic approach (this is, in my opinion, also the reason why he has been often considered the “founder” of Viśiṣṭādvaita Vedānta —a term which he and Yāmuna still ignore). As for Nāthamuni, his relation to Vedānta can only be presupposed out of what we know of him through his successors, given that his works have been lost. Their titles focus, however, on Nyāya and Yoga (and not on Vedānta).

As for No. 2, we know nothing about Nāthamuni’s relation to Mīmāṃsā, but we know that at least one trend within Vedānta (as testified by Śaṅkara’s commentary on the Brahmasūtra) claimed that the study of Pūrva Mīmāṃsā was not necessary. Yāmuna’s relation to Pūrva Mīmāṃsā is double-faced, but Pūrva Mīmāṃsā authors seem to be his targeted objectors in the sense that he wants to convince them of the legitimacy of the Pāñcarātra transmission (although often recurring to their same arguments) and he by and large adopts Nyāya strategies (such as the reference to God as the authoritative source of the epistemologic validity of the Pāñcarātra, or the use of inference to establish God’s existence). The situation changes, perhaps during Yāmuna’s own life, certainly with Rāmānuja, who steers in direction Vedānta and, thus, comes closer to the Pūrva Mīmāṃsā. So close that he programmatically states at the beginning of his commentary on the Brahmasūtra that not only the Brāhmaṇa part of the Veda needs to be studied, but that its study is part of the same teaching with the Vedānta. Veṅkaṭanātha takes advantage of this (perhaps causal) remark and understands its deep implications: he can thus state that the whole of Pūrva Mīmāṃsā and the whole of Uttara Mīmāṃsā constitute a single teaching (ekaśāstra).

No. 3 and 4: Furthermore, Veṅkaṭanātha used the same model, I think, to incorporate into the system further elements. He reaches back to the Pāñcarātra, which had been defended by Yāmuna but rather neglected by Rāmānuja and, more strikingly, to the hymns of the Āḻvārs. It is in this sense more than telling that Veṅkaṭanātha (as first among the first teachers of Viśiṣṭādvaita Vedānta) decided to write also in Tamil and to write theology also in poetical form, as the Āḻvārs had done.

Did you ever try to reconstruct what had happened in a single school within some generations of teachers and pupils? What did you find out?

On Yāmuna’s role in what came to be known as Viśiṣṭādvaita Vedānta, check this post, on Pāñcarātra and Vedānta, see this post. For more on Veṅkaṭanātha, check this tag.