if yes - then Bhagavad Gita would say that you shouldn't convert to another sect. I am asking this question in regard to the Vaishnavite temple a smarta guy opened in New York state. To me it is jarringly "in your face" given how many great, established eclectic Hindu temples exist in the area. Did he feel all this life that he was a vaishnavite trapped in a Smarta body/mind/soul?
"Swa" means "own". And Dharma can mean many things like mentioned in this answer.
It can mean religion, duty or even tendency.
Now, if someone is born into a particular family then certainly he gets bound by that family's traditions and customs. In other words, his family traditions are his "own traditions".
So, to fulfill those duties is also mandatory for him, besides fulfilling the duties that are attached to him due his caste (Varna) and the stage of life (Ashrama).
See the following passage from the Devi Bhagavata Purana:
14-24. Right way of living is the best of all the Dharmas and is great Tapasyâ (asceticism). The knowledge comes from this Right Living. Everything is attained thereby. He who is devoid of Sadâchâra, is like a S’ûdra, even if he comes of a Brâhmin family. There is no distinction whatsoever betwen him and a S’ûdra. Right conduct is of two kinds :-- (1) as dictated by the S’âstras, (2) as dictated by the popular custom (Laukika). Both these methods should be observed by him who wants welfare for his Self. He is not to forsake one of them. O Muni! The village Dharma, the Dharma of one's own caste, the Dharma of one's own family and the Dharma of one's own country all should be observed by men. Never, never he is to do anything otherwise.
From the Purana's 11th book's 1st chapter, verses 14-24.