Your UT CFP is invalid in MI or any other State that does not accept non-resident CCWs the moment you cease to be a UT resident. In your case, you need to get a MI CPL the moment you become a MI resident as evidenced by possessing a MI DL and having your personally owned vehicles registered in MI. You can and I encourage you to renew your UT CFP as a non-resident UT CFP and have it for traveling purposes. It's inexpensive to maintain and one of the preferred CCWs; resident or non-resident. Be sure to notify UT BCI of your change of address. I also encourage you to mail off your UT renewal a little before the 60 days prior to expiration. One of the reasons I get so much business as a traveling CCW instructor is because I teach the UT class.
There is no grace period in MI. The only States that have a grace period that have a resident CCW requirement are CO (90 days), FL (90 days), NV (60 days) and TN (180 days). You also need to be a MI resident for six months prior to applying for a MI CPL. The lack of grace period and six month residency requirements are two reasons I would not entertain moving there.
Not all States have a requirement that residents must have their respective CCW; IIRC, the list of States that require their residents have their respective CCW are AL, AR (administrative requirement by AR State Police, no statutory requirement), AZ, CO, DE, FL, GA, IN, KS, MI, NH, NV, SC, SD, TN, WA and WV; less than half of the States with recognition or reciprocity. In all other States with recognition or reciprocity, residents can legally carry on a non-resident CCW. It's not a bad practice to have back up CCWs in case yours accidentally lapses or is lost. This is something to keep in mind when considering relocation if legal CCW is a high priority for you.