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The non-profit can still have rules in place on who is accepted as members and who isn't, as long as the rules don't break any laws (discrimination etc.). They also can have a mission statement and rules that are not up for vote. They can have members enter a contract with the non-profit when joining as a member.

Also, depending on who's "taking over" and who's "taken over", it can be argued that if the majority of members that provide the majority of the funding to the non-profit, shaping the non-profit according to the majority of its members is the right thing to do. If there's too much differences in opinion, the non-profit can also split any time.

Indeed this can (and does happen) but the non-profit can loose its status if it starts doing stuff that does not further its purpose as stated by the bylaws. And the tax authorities can decide that your stated purpose would not be tax exempt (for example a union would not be tax exempt and neither would a collector society [think for example a numismatic society])

This is in fact possible and done sometimes, if the membership rules allow people to join like that.