Just some random thoughts, not meant to be definitive:
The concession on Divorce is found at Deut. 24. All of Deuteronomy up to chapter 31 is one long discourse comprising "the words that Moses spoke to all Israel in the region of the Jordan in the wilderness." (1:1)
Of course, the Pharisees, whose religious descendants became the rabbis of today, considered as torah, or "law," both the written text and its interpretation (the "oral law").
Jesus accepted the Pharisees as those sitting "in Moses' seat" in his day. It was their deeds and traditions he objected to, not necessarily their core doctrines. Paul, another Pharisee, also contributed to the Christian corpus of doctrines. In a sense, both rabbinic Judaism and earliest Christianity were formed from a Pharisaic matrix.
Perhaps Jesus' comments merely suggest that a plain reading of Genesis makes it clear that "what God has joined together, let no man put apart." Therefore, the allowance of divorce, as promulgated by the Mosaic Law at Deut. 24, had to be a concession.
Moses was known to wrangle with God at times for the benefit of his (and His) stiff-necked people. Perhaps this is one of the things, though not specifically mentioned, that Moses wranged from the Eternal?