Will Migration Assistant transfer 32 bit apps even though they won't work

Will Migration Assistant transfer 32 bit apps

Yes and no. As I recall from other's experience the 32 bit apps are included but are moved to a folder called Incompatible Software. I deleted all of mine before upgrading from Mojave to Big Sur in 2020 and didn't experience this myself.

Regardless of that, 32 bit applications are useless in Monterey and you should remove them from you old Mac before you do a final Time Machine backup, which you intend to use to set up your new Mac.

There is another school of thought, as you can read in a recent thread here, that you should move only your Mac user account, and then reinstall all the content manually. Read the post by P. Phillips which is very detailed.

Would suggest following the previous advise by David MacKinlay and remove the 32 bit application before making the move to Monterey or doing the final TM Backup before attempting the Monterey UpGrade.

Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina, Big Sur & Monterey supports only 64 bit and there is no workaround.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may required update or upGrades to 64 Bit Versions.

Any 32bit app that is moved will have a line through it in the applications folder which you can delete later as desired.