After the Third Council, the Vibhajjavādins gradually evolved into four groups: the Mahīśāsaka, Kāśyapīya, Dharmaguptaka and the Tāmraparnīya. Theravada is descended from the Tāmraparnīya, which means 'the Sri Lankan lineage'. On the other hand, some sources suggest that Mahīśāsaka, Kāśyapīya and Dharmaguptaka did not evolve directly from the Vibhajjavādins, although an original connection between these groups is posited due to the similarities of their respective Vinayas.
The Vibhajjavadins are claimed to have seen themselves as orthodox Sthaviras.
According to Sinhalese tradition, Buddhism under the name of Vibhajjavāda was brought to Sri Lanka by Mahinda, who is believed to be the son of Emperor Asoka, an event dated by modern scholars to 246 BCE.
The Theravāda descendants of this school claim that Vibhajjavāda represents doctrinal orthodoxy. However, proponents of this approach are frequently mentioned in the Sarvāstivādin Mahā-vibhāsa, where they are viewed as the type of heretics who "make objections, who uphold harmful doctrines and attack those who follow the authentic Dharma". This characterization was a response to the methodology and type of arguments recorded in the Kathavatthu, where part of Sarvāstivādin doctrines is refuted. Only three questions in the Kathavatthu are directed against Sarvāstivāda, which probably means that the rest of the Sarvāstivāda doctrine did not differ much from the Vibhajjavādins at the time of the Third Council.
Lance Cousins, "On the Vibhajjavādins: The Mahimsasaka, Dhammaguttaka, Kassapiya and Tambapanniya brances of the ancient Theriyas", Buddhist Studies Review 18, 2 (2001)
Prasad, Chandra Shekhar, "Theravada and Vibhajjavada: A Critical Study of the Two Appellations"' East & West Vol 22 (1972)