I wouldn't concern myself over that at all. Moving less frequently does not necessarily equate to happier in a branch or in a career field. What will make him happy is serving in that particular branch. A happy AD member, makes a happy family member.
If you are looking at pure stats, than I would have to say the AF. The AF is known as the "corporate" branch.
Bullet served 21 yrs. We moved on the avg @30 month marker. Our shortest tour was at FTU (6 months), our longest was in UK at 38 months. We had friends that moved a total of 4 times, and because they didn't want to move the kids, their spouses took 1 yr remote tours. A close friend that we started off with as 1st LTs. Moved to NC at the same time as us in 98. They never moved again. They retired the same time as us in 08. We however moved in 01, 02, and 05 and moved to VA in 08. My kids attended no less than 9 public schools from K-12. The two oldest both moved 2 times mid-school yr. Their kids attended 4 schools, they for all purposes would be considered townies since from pre-school to HS they grew up with the same kids. The 4 schools, they attended were all in Goldsboro, NC.
As you can see, you just can't make a judgement call on which one moves less or deploys more because even in the same community, everybody's path can be different. A lot has to do with their career/promotion path also for the AF. That 1 yr marker for us was when Bullet went to CGSC as an O4. He also decided to switch out of the 111 to the 15E which meant he had to do 6 months xtraining. 6 months + is considered a move. He did these things to have a better chance of being promoted along the way. We know several officers that didn't want to move, one was an AFA grad. In the end when the O5 board came out, they were all passed over and forced to retire as an O4. They opted to put family life in front of the AF's needs, and the AF reacted to their decision by passing them over.
Every branch has their family support system for when the member's are deployed. Although, we were never with the Navy for jt assignments, I would have to say between the Army and the AF, hands down goes to the Army. They are really strong in this area, but my assumption is they are because they have to be due to the lengths of their deployments. That and the fact that most of their TDY's are in tents in the field, whereas, the AF members are usually in hotels, and can get a phone signal or skype access easier!
As far as on base housing, AF wins that hands down. Every yr the branches rank the best housing, and AF usually comes out the top. However, in the defense of other branches, the AF is the smallest, thus, it is easy to maintain those houses just due to the amount of houses.
I am willing to bet that most people on here will say their branch is the best, but if you look at really why they say that, it comes down to the fact that they love their branch, they want to be in that branch and not any other branch. Thus, moving 10 times in 20 yrs is not a problem, it is not a negative. They see it as a positive. I didn't want to live in Idaho, but if I didn't I wouldn't ever have seen how beautiful Boise is. I wasn't thrilled of moving to Leavenworth, but I was glad we weren't going to Alabama and loved Kansas City. It pained me to see my DD cry everytime we moved, but now that they are young adults their bond is closer than most because only they share those memories. Their friends think it is so cool that they lived in AK, and that they were born in the UK. It is all how you see it.