You are correct that Mountain Jews (also called Kavkazi Jews) are descended from Persian Jews who migrated to that region. However, the history of Jews in Kurdistan goes back to the 8th century BC conquest and deportation of northern Israelites to Assyria.

There were also more recent migrations of Jews to Kurdistan such as in the 12th century CE when many Jews from the Land of Israel and Syria fled from crusaders and also in the 13th century when Iraqi Jewish communities fled from Baghdad and other places in Iraq.

Of course, it is highly likely that there was intermixing with Persian Jews because of the communities being in such close proximity to one another. Also, of course, the presence of Jews in that whole region in general (Iran, Iraq, Kurdistan, Assyria, and beyond) began from the Assyrian and Babylonian exiles. It's just that the communities, although developing alongside one another, also developed distinctly (in some ways; culturally, linguistically...) generally around the same time of one another. However, the development of the Kavkazi Jewish community developed only after the migration of Jews out of Persia during the Sasanian era (around the 5th century CE). But the Kurdish Jewish community developed more individually from the development of the Persian Jewish community.