The following states honor the MI permit:
AK*, AL, AR, AZ, CO, DE, FL, GA, IA, ID, IN, KS, KY, LA, MN, MO, MS, MT, NC, NH, ND, NE, NM, NV, OK, OH, PA, SC, SD, TN, TX, UT, VA, VT*, WA, WV, WY
Traveling through states that don't honor your permit, the easiest way to transport the gun is in accordance with FOPA, 18 USC 926A:
§ 926A. Interstate transportation of firearms
Notwithstanding any other provision of any law or any rule or regulation of a State or any political subdivision thereof, any person who is not otherwise prohibited by this chapter from transporting, shipping, or receiving a firearm shall be entitled to transport a firearm for any lawful purpose from any place where he may lawfully possess and carry such firearm to any other place where he may lawfully possess and carry such firearm if, during such transportation the firearm is unloaded, and neither the firearm nor any ammunition being transported is readily accessible or is directly accessible from the passenger compartment of such transporting vehicle: Provided, That in the case of a vehicle without a compartment separate from the driver’s compartment the firearm or ammunition shall be contained in a locked container other than the glove compartment or console.