… but what are issues I might encounter?
I’d like to preserve all my /usr/home files and all the packages I have previously installed.
One PC has the operating system in a separate partition, while the other, a laptop, is setup as one partition for the OS and /home together.
How would you go about this?
Were it me, I would…
NOT try to copy the existing 22.04 MATE system.
Install a clean Ubuntu 24.04 Desktop (Gnome) instead. It is faster and easier.
NOT try to copy any packages.
Install those applications fresh from the 24.04 App Center.
Deb packages are often incompatible with different releases, a common cause of problems.
Do copy your data from your /home/$USERNAME directory (not /usr/home). Not the whole directory. Just the data and settings you really want.
Most systems won’t have a /usr/home/ directory; but you likely mean $HOME or /home/guiverc/ for my own user files, but I’m not sure, as /home/ and /usr/ are different directories that can be treated differently.
Ubuntu 22.04 LTS used the ubiquity installer, which allowed an ~easy non-destructive re-install option, which allowed the preserving of files, and even auto-reinstalling the deb packages that were installed from Ubuntu repositories; however both Ubuntu 24.04 LTS and Ubuntu-MATE 24.04 LTS use the ubuntu-desktop-installer where problems were discovered in ubuntu-desktop-provision late in the development cycle; thus a forced format was used as a workaround which prevents the non-destructive re-install.
Maybe of note: The non-destructive re-install I mention worked by taking note of what packages you had installed; erasing the contents of system directories; then doing the install, before finally downloading & installing the additional packages you had installed - ie. it did NOT use your prior binaries, but installed new ones for your new release, thus complying with what Ian already suggested.
The problem I just mentioned will be very noticeable on your single partition (laptop) system, as a re-install will force your to lose your /home directory, as its only possible to keep it IF on a separate partition with 24.04 & 24.10 currently where ubuntu-desktop-installer is used; only the calamares installer on those releases won’t force a format. The result of this is you’ll not only not get your prior installed packages re-installed; but a forced-format will mean your data also doesn’t survive (if on the same partition as /). You can re-install on your desktop & have your data survive.
I wasn’t sure what you mean by try to migrate; as to me it was you attempting to replace your two existing systems with a new single system & thus wanting to merge data from two sources into a single system which adds more complexity, and may require knowledge of what packages/apps you’re actually using, so I ignore that detail.
Ian’s prior advice was good !
Yes, I really meant $HOME, I had a brain fart 
I wonder if I could use gparted to shrink my partition, create a separate partition to install 24…04 Ubuntu Gnome, then remount /home to the old shrunken partition?
Would this result in me having my old Mate OS and my new GNOME using the same $HOME???
Yes you can share a /home/$USER/ directory between two different GNU/Linux operating systems, but are you aware of the pitfalls of doing that? especially when the software is different versions? (ie. you’re mentioning Ubuntu [MATE] 22.04 LTS and Ubuntu 24.04 LTS meaning software versions will be ~two years apart!)
More than a decade ago I used to share $HOME between my dual boot systems, but stopped doing it after problems & data loss.
If your data is important to you, consider the potential pitfalls, and do your homework, by comparing each app & the changes made between those releases, especially where database changes, new features etc have happened, and what impact that will make to your datafiles, as the older/newer version may treat those differences in data rather differently given its may not be aware of changes (esp. if made by a future/newer version of the software) or may not expect data to be written that way.
I’d not recommend sharing $HOME between OSes of different releases.
Sharing home itself has been problematic at best for as long as I’ve messed with *nix. However I’ve gotten around this by having a separate data partition specifically. For me this is an essential part of being able to distro hop at with little real risk. This data partition has a folder for each user with the usual suspects… Documents, Downloads, Pictures, Videos, Music + whatever others one may want to share between distros. Stuff like .steam, .wine and so forth.
I just symlink most of the data folders into /home/“$USER”. I bind mount others such as /home/“$USER”/.var (flatpak) because it doesn’t follow symlinks. It works quite well if you are the primary or only user on the system. Then you can have as many /home directories as you want with no fear of collisions between distributions and configuration files. Obviously it can get messy when your system has more than a few users. One could probably automate this with a script calling the adduser command but that would likely be very specific to each system as everyone would have a different method of going about it.