This is literally how Islam got to places in Asia and spread. It was through Sufis who acted as initial points of contact.
> A lot of these traditions are much more flexible than whatever is the dogma or institutional version of the time suggests.
This is only so when it comes to missionaries in each religion. Be it Evangelists saying Vishnu is Avatar of Christ (we literally have Christ idols shaped like Vishnu with multiple arms and sitting on a lotus in a Church in India), Sufis singing Krishna is same as Allah or heck even ISKCON claiming Christ is an Avatar of Krishna /Vishnu. That does not make "traditions flexible". It's just a cheap tactic to increase followers/harvest souls. These converts are typically fluid in their own convictions and can easily be swayed one way or the other based on reasons unconnected to the Religion/Dogma/Philosophy itself. Reasons could range from political, monetary to finding fame/popularity within a new sect/group/cult/ideology.
Very few are seekers of Truth.