Mark 4 bases and badger rings

Are you referring to picatini style, if so that is a standard size and should work. I have that base and use several different rings on it over time.
 
Should work fine. I have done the opposite many times. I have Badger rails on several rifles and quite frequently switch scopes with either NF, Badger or MK IV rings already attached. They always mount up just fine.
 
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