Svami Desikan did not have a single disciple who propagated a single tradition. He had many disciples and through them, the tradition has spread widely.
This did not apply to Desikan alone; Upto Ramanuja’s time, there was a single tradition until his period. Ramanuja had many disciples. He anointed 74 Acharya-s. As a result, there are 74 lineages of followers of the tradition. Among these followers, there could be minor differences of opinion in messages, ritualistic practices, etc. Such differences become a tradition on their own. These are permitted by the scriptures; Svami Desikan himself has said this – ‘Due to differences of place and time, scriptures are agreeable to differences in ritualistic practices.
Scriptures have accepted that not all places can follow the same code of practices and there could be differences due to place and time. The important message is that Sriman Narayanan is the Supreme Truth; we must surrender unto Him; among the followers of Svami Desikan tradition, none of us differ in this aspect.