OK, a quick update for those who are interested...
After some experimentation, here's what I've learned:
The "Save Ultrabeat Samples with Project" setting in Assets does resolve much of my issue, seemingly. So thanks again for that suggestion.
A couple of caveats:
This is a problem in MainStage, as there is apparently no such "save assets" option in that program. This is where I originally created the kit and sequences, because it was intended for a live performance. (Which is of course why I never even thought about the "Assets" option, as it doesn't exist in a MainStage Concert as far as I can tell.)
So what I did was to go back and create a new Logic project, created a track with UltraBeat, recreated the kit, saved the setting, and then saved the project with the appropriate "Assets" options checked.
Then I went back and created a new Concert in Mainstage, created a channel strip with UltraBeat, loaded my setting and, VOILA!, all was well.
HOWEVER, when I move this to my MacPro, not only do I have to move the UltraBeat Setting file, but I have to copy and move the folder from my Logic project with the UB samples in them, too. (Obviously)
But then when I try to load the setting in UltraBeat on the MacPro, I get the same message as before, asking me to find a sample (just one) from the kit. At this point, I navigate to the folder with said samples, find the one requested, and hit enter. From there, UltraBeat seems to be able to find ALL the required samples and loads them up.
So this is certainly workable, and again I appreciate your help. However, I'm wondering if there is a location I can put the transported samples such that Ultrabeat will find them without asking when I move between computers...?
Anyway, thanks again. And a shout out to Steve, Igor, and everyone in the community. You guys are great.