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What happens if one of the upgrades on one of the two characters has a Put on only restriction that the other character doesnt meet?

In that case, that upgrade stays where it is. If the other character has a upgrade of the same type, it also stays where it is. Other types of upgrades may still be swapped.
Source: Wizards of the Coast (2019-03-01)    【Link to Ruling】
Tags: Private Red Heat, Infantry • Emergency Response