I don't know much more than the general bit about the Brits bringing Freemasonry to the States in traveling military lodges back in the 1700s. Personally I didn't join the craft till after I was done with my tour.
There are some Brethren about though that are currently serving in military lodges that hopefully can chime in. I think though that each of them get a dispensation from different places and all have their own unique story.
Personally, when a military friend of mine asks me about Freemasonry I usually urge them to join the Fraternity under the auspices of a Grand Lodge while they are abroad. Reason being that often those Grand Lodges have relations with both main Grand Lodges in each of the states in the US, meaning they have no barrier to fraternal visitation to either. One more way to thread the needle.
I believe that getting together, in person, will do a whole lot. Breaking bread together. Serving in degrees together. Heck, BBQing ribs together. In person, face to face, getting to know each others families. Its why I really want visitation to get fixed up here in Texas.
Another thing - the Shrine has talked about removing the Master Mason's requirement some recently. Really, I hate the idea - but I will say that, in Texas, it might have a side benefit of maybe, just maybe, being a place where GLoTX and PH folks could come together under one roof for well and true. But that is a whole can of worms for a fact with good arguments on both sides.