Based on DNA, it appears most Ashkenazim originated in the Middle East. There were many converts in ancient times, but apparently more female than male converts (that might be due to the loss of civil rights a male would have in converting, as well as bris), but not all that many converts since the rise of Christianity (which made conversion to Judaism a capital offense). We have enough Middle Eastern ancestry that Europeans have always perceived Jews as looking like people from the middle east (suggesting the rate of conversion wasn’t very high once we arrived in Europe). The Sefardim also had some converts (again , before the rise of Chiristianity and Islam), but they would have been largely middle eastern looking.
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