Even the very issue of whether or not in ancient times bhikkhu/nis could / would travel and practice among a range of different "sects" / "traditions" / "lineages" or not is in dispute.
Many have argued that they did not. But often such arguments have been based on heavily sectarian literature. More broader research suggests that this practice was very common.
In india, I wonder how much Brahmanic ideas of caste influenced such things, the idea of purity through gotra and kula, the idea of purity through ritual - transferred to purity of ordination lineage and purity through ordination rite. I'm going to finish off Bronkhorst's book before going further with this, but the more I look beyond the Buddhist tradition alone, the more I think that there is something in this on a number of fronts.
My recently moved Blog, containing some of my writings on the Buddha Dhamma, as well as a number of translations from classical Buddhist texts and modern authors, liturgy, etc.: .