What does the Tanach say on How non bani israelites embraced the religion of Prophet Moses at the time of Prophet Moses.
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The TaNa"Ch is not at all clear on this subject.
There are examples, however, of non-Israelites who joined Bnei Yisrael (ie., converted) in the times of Moshe, including the 'Erev Rav and Moshe's own father-in-law Yithro, as well as Moshe's wife Tzipporah, for that matter, but there is no real prescriptive approach to converting given. On the other hand, Yithro is a subject of dispute as to whether and when he converted.