Tested this a little bit today. Currently Mythril Wings of Service (lost two medals from switching nations).
First I tested if it increased the "time cap". At this medal, the cap should be 86 exp/AN per minute. I went to a defensive campaign, got tags, spammed spells and redeemed tags after about 90 seconds, and was awarded 85 exp / 86 AN, so no change there.
Then I tested if it increased the base points awarded for actions. I went to an offensive area without the moghancement, forced an HP change of exactly 200 HP and recovered to full with Magic Fruit. I received 10 exp/ 5 AN (this is also interesting because apparently curing is 5% now, changed from the previous 10%). Then I changed to Moghancement: Campaign and returned, and did the same thing, again receiving 10 exp / 5 AN. So no change there either.
It could possibly be that it only affects tags that you've had for a certain duration, or only certain classes of actions, but at this point I have another theory. The text reads, "Moogle-powered assistance from your Moghouse. Improves your evaluation for services rendered to the Allied Forces." In the game's terminology, "assessment" is used for campaign exp/AN awards, while "evaluation" is used for rank up, so what I'd surmise is that there is a hidden point value awarded for each campaign towards retaining/improving your medal, and that the moghancement gives a multiplier to that value. I considered testing that, but since it's not even nailed down (to my knowledge) what the base requirements are for rank-up, I don't think I'll bother, and just set the moghancement anyway whenever I'm campaigning to rank up or keep a medal.