You can transport and firearms that you can legally posses in your home state anywhere in the USA. This is governed by the ICC. I went through this as I was taking gunsmithing classes in NC and traveling from MA. Called ATF and they referred me to a gal at ICC. As long as they are locked up and totally out of your control, you are good to go. Glove compartment or center console, even if locked is not considered “ out of your control” . To be safe, ammo in a separate locked case.
Now, that said, some states you can not leave firearms unattended in your car as you go to lunch etc (NJ)....Some states, if you are shooting in state, you need proof you are attending an organized shoot at a shooting range certified by the state. (NJ)...some states you can not have your guns in a hotel/motel overnight. They might have to be stored at the range. As far as I know, these draconian laws only apply to Maryland , California and a few New England states. Once out of this God forsaken part of the country, you are mostly good to go.
In NY, you can not be in state with your guns more than 48 hours before the match and need to get guns out of state within 48 hours after the match.
There is confusion on this point. Need to check with local LEO . You can not violate local laws on magazine count, assault weapons, automatic weapons. Guns must comply to local law.
No license is needed to transport firearms anywhere in USA. The ICC lady commented that the truck driver that carries a load of handguns from S&W in Massachusetts, through NY and NJ, most likely does not have a license for all those guns. Good point!