How does dual-military work whenever the spouses are in different military branches?

USAF and USMC are not colocated anywhere below COCOM level, where you would not work until at least CPT or MAJ, and both of you getting the same at the same time would be unlikely.
The closest the USAF and USMC get are Guam, Okinawa and Hawaii, then:
Joint Base Andrews and Quantico at 42 miles.
The odds of getting any of those combinations at the same time are very slim.

You two should seriously consider joining the same branch, so that you have a formal program for getting stationed together.

Throughout my entire career, I only had two assignments where I was stationed with Marines (USCENTCOM/HQ and Misawa AB Japan). Assignments together could be possible but it will severely limit your options. My suggestion is to pick a branch you can both be happy in and commission there. The odds of you being stationed together would be much higher.

I’m just wondering you are in the army NG already why switch branches and hit stay army? Why you two won’t commission in the Same branch? It will be hard due especially if you can’t find a base near each other cause how can you live together in you not even stationed together?

You would have much better luck commissioning in the Army as there are Army/Air force bases that are co-located. If you go Marines and your partner goes Air Force, enjoy taking leave to see each other or you are going to have one hell of a marriage.

Each branch will employ its service member IAW the needs of the service. Different branches don't have married couple programs. Couples in the SAME service, yes, but not different ones.

As others in this string have said, this will be easier if you’re both in the same Service. The Army and Air Force have several Joint Bases where Airmen and Soldiers work reasonably close to each other. Navy and Marines also have bases where Sailors and Marines work in shared facilities
As you try to sort this out, you both need to get into “universally assignable” specialities that are used at most installations. For example, transportation, supply, logistics, medical, personnel, finance, contracting, information technology.

You tagged marriage. Did you go to the actual link? There are resources there for each branch and Joint from the Army perspective. https://www.rallypoint.com/topics/marriage/about