Suggestions for taking photos during fast moving tours?

I’ve “rediscovered” photography with my mirrorless digital Fuji camera in retirement.  When my husband and I are traveling on a tour, I’m discovering that I’m struggling to often get good well-composed pictures when the tour is moving pretty quickly.  If I have time afterwards or on my own, I can take the time to set things up and get better shots.   If I try to take my time — I often hold up the whole group (no, these are not photography tours — general interest).   I often find myself using the phone as it is convenient and faster and have even taken an iPhone photography course to get better using it.    
 
Any suggestions how to handle quick moving guided tours with the Mirrorless DSLR?  I tend to shoot Aperture priority and have my basic settings set up before I embark on the tour.    What do other people do in those situations?  
 
Very different than when on one’s own with lots of time…..or a purposeful photoshoot…..
 
Thanks!

Imported from legacy forum.