How does inprocessing work for a married soldier?

As long as the assignment is command sponsored, you'll be treated the same regarding your entitlements. It doesn't matter if your husband is physically with you or not to the government. In other words, you'll still be able to get on the housing list, etc.

Keep in mind that from the government's view, it's about entitlements. However, if you are 'geo-bacheloretting', you may want to discussion with your unit leadership 'what the art of the possible' is regarding your situation. For example, you may not want to stay off-base until your husband arrives because he's going to help with looking for a place. Depending on the policies in place, you may or may not be able to stay in the barracks if there is space available until he arrives

Obviously things that require your husband's physical presence will have to be done at a later date, but "paperwork drills" can all be done. One suggestion is that you get a limited power of attorney so you can legally sign for him in situations you two decide that you should be able to take care of (for instance, if you were required to have both of your signatures on a housing lease - without it you'd have to wait until he arrived).

Since this is your first OCONUS trip, HIGHLY recommend that you check out a lot of the Military OneSource* guides regarding preparing for an OCONUS move as well as living overseas. This is specially true if you've never lived overseas before as there are many things that will 'bite you in the backside' the first time you experience them (What, you mean electricity is different in Germany?
What do you mean my cell phone doesn't work here? Who do I call if my car breaks down in Korea? etc.) A few additional tips:
- Levy Briefing. Have your husband attend this with you if possible. Depending on where you're stationed, some posts have in-person briefings while others have moved it all online. Regardless, make sure you go through the briefing and ask questions (in-person or to the POC given if online).
- Household goods (HHGs). Housing overseas is typically a lot smaller than what you're used to in the US. If you don't have a lot of stuff (or you live in a pretty small place right now), you might not have to worry about where it goes. However, if you have accumulated the typical stuff we drag around with us in the life, you might want to think about what you can get rid of or put into long-term storage.
Additionally, whatever you're having shipped as HHG might take MONTHS before you see it again - so obviously you want to keep that in mind when planning.
- Driving. Make sure you're driver's license isn't going to expire while you're over there (renew it early if so) and make sure you check with your insurance company about coverage you'll have overseas. Check out USAA* (even if you don't have them as your insurance company) for some important country specific information. Study up ahead of time for the test in country and take it soon after arriving.
- Sponsor. Develop a great relationship with your sponsor - they will be your lifeline until you get your feet on the ground regarding what to expect while you're there. ESPECIALLY regarding local policies that will impact you.

Also, don't forget to check out the welcome page for wherever you are going.
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* https://www.militaryonesource.mil/housing-life/living-abroad/ (examine the MilLife Guides, Resources, Articles, etc.)
* https://www.usaa.com/inet/wc/ins_intl_coverage_landing_mkt?vurl=vurl_internationalpcs&

Each Garrisons requirements are based on resources, and how soon your Family is expected to arrive. You will likely be put into the Barracks until your Family are ready to fly in. Once they are verified and housing is available you will then transition to housing, on or off base, again based on Availability.

In processing is as smooth, if not smoother than mainland bases. The key is to make sure your Families eligibility to join you, i.e. Family care plan as well as medical screening.

You should reach out for a sponsor and contact the i processing center for your gaining installation.

If you know what unit you are going to then inquire if you have a sponsor. I told my units all incoming soldiers will have sponsors. They make the transition infinitely easier.