In Missouri you get a "Permit to Purchase" from the sheriff of the county where you reside. The fee (tax) and wait varies with each sheriff - what this boils down to is that the sheriff's of the urban counties make you wait for the permit for the maximum 7 days allowed by statute, but in some rural counties you walk out with the permit the same day you turn in the paperwork. The biggest problem with this is traveling to the courthouse to get an application; it's not a problem for me here, but where I grew up in the Ozarks it means a 26 mile trip one way (on the other hand, everyone knows me, so the permit will fly right through and negate the need for a return trip).
You do not need a serial number or description of the weapon you want to buy, and you have 30 days to exercise the permit. After the purchase goes through, there is another form that is returned to the sheriff's office with details about the exact firearm; in essence, it has become "registered" in the county records. I don't know if this is permitted or required by the statute, or if individual sheriffs have taken it upon themselves to take this step. If you want to buy 3 handguns at the same moment, you need to fill out three applications and pay the tax three times.
The application in St. Charles County will quiz you about your recent residences and how long you have lived in the county, plus info that is also on the 4473.
When you move here, bring all your firearms along - the state doesn' t want to know about them.
What part of the state will you be moving to?