I think this 'local religion' bit is overplayed. Christianity, Islam, Buddhism, Hinduism are world religions which have persisted long after any political entity which may of carried them has ceased to exist.
No - you are missing the point.
Of course Christianity, Islam, Buddhism, Hinduism have been carried by believers from one place to another and those believer have often attempted to convert the people they have found either through persuasion or force.
My point is that, as far as I know, no-one has ever become a Christian, a Muslim, a Buddhist, a Hindu etc without being made aware of that religion from another human source. If these religions are about divine revelation, that wouldn't need to be the case - the Christian god could make himself known to a remote tribe in the Amazon and teach them that Jesus is his son etc etc (just as the bible suggests the christian god did to many people in biblical time) - so you'd get a tribe that
independently became christians (muslims, hindu etc). But it never happens.