I know that the groomers that work at vets hate the rest of us groomers for this, but SO many groomers, including myself, send the very elderly, frail, and very naughty dogs to the vets for grooming. I have referred many very old dogs, obese dogs, and mean dogs to vets..(and I used to work at a vets too...so I know what its like to have to do those dogs...) I tell them that it simply isn't safe for me to groom their pet anymore. They don't get a choice about rescheduling...I just give them a referral, and a have a good day. There are few old dogs that I have to send off though, if I have been grooming them for years. I talk to the owner, and we discuss a cut that is easy and fast, and that will last a long time, rather than stress the old dog with another facility, groomers, etc. (My favorite, a 7F everything on the haircut dogs, no clean feet on poodles anymore, etc.)
Cats are a different story...it is very dangerous to groom an elderly cat..heck, even a young, healthy cat can have a heart attack and die instantly, under minimal stress. I have clients with older pets sign a release form, so that they acknowledge the dangers, and if there is any sign of stress, the cat is put back in its carrier and owners called immediately. People have to understand the stresses grooming has on those old dogs and take steps to alleviate it as much as possible.