Ok, here's the basic facts.
First a question: When will you be moving to Michigan. It's going to make a difference as there is a change in the law in January.
As of RIGHT NOW, until January, here is the current law.
You will need to get a "Permit to Purchase" from your local PD for each handgun you bring into Michigan.
A handgun does include any rifle between 26" and 30" inches with the stock folded or collapses. (Below 26" is a SBR and they are not allowed)
You will need to go the police station and get and fill out a "Permit to Purchase" for each handgun and return to the PD to get a "Safety Inspection Certificate" for each handgun. (Sometimes you go the Sheriff's Dept instead, but only if there is no PD for where you live. The Sheriff will not issue the permit purchase or do the safety inspection if you have a local PD that provides this service)
Since you are moving from out of state you might be able to do the "Permit to Purchase" and "Safety Inspection" on the same trip. That is because you already have the guns in your possession and this is really set up for people buying new guns they don't already possess.
You do need to do this as compliance is required by law.
Now, if you move AFTER January there is a change in the law. After a certain date in January (which I can't recall off hand) you will no longer need to bring the handguns to the PD for the "Safety Inspection."
Instead you will still need to get a "Permit to Purchase" and then, instead of bringing them to the PD, you will just fill out the paperwork itself and mail it to the State Police. It is still required by law and they actually call it "registration" now. Like I said though, that doesn't change until January, so it depends on when you move.
If you move before January do NOT just wait until January to take care of this as the law is vague as to how much time you have and you do NOT want get in trouble for noncompliance. It has happened.
Now for the Michigan Concealed Pistol License. The bad news is you can not apply until you have been a Michigan resident for six months. There is no way around that and no way to "transfer" an out of state permit into a Michigan CPL.
You will also need to take a class in Michigan, that meets specific Michigan law, in order to apply for your Michigan CPL. You can take the class before you are a resident, but can not apply until you have been a resident for six months.
The class is easy to find and there a bunch of instructors all over the state. (Especially in the Metro Detroit area for some reason)
You will need to take the class in Michigan though as there are certain specific things that have to be included and there is certain wording that has to be added to the certificates. Out of state classes do not generally discuss Michigan law specifically and do not include the Michigan required wording on the certs. (Except for a couple instructors in Indiana right by the border)
The really bad news is that there is no legal way for you to carry a concealed pistol in Michigan until your six months are up and you apply for a Michigan Concealed Pistol License.
Your Arizona permit is no longer valid as soon as you become a Michigan resident. We only recognize out of state permits as long as you are a resident of that state. That means you can't carry on say a Florida permit or anything either.
And, since you need to wait six months to apply for a Mi CPL, it is to your advantage to declare Michigan residency as soon as you cross the border to get that six month clock ticking asap.
Any questions?
. I recently read that the Mandatory handgun safety inspection has been repealed. And I plan on obtaining a Michigan CWP as soon as possible after I get there. Should I try and get it before I move? Do I have any grace period to get it? 
