Let me offer you some support on a couple of different levels.

First off I appreciate your concern about moving the animal that is about to vomit as you don't want to accidently cause the animal to aspirate any vomitus. If you are able to scoop your hand under the back feet with one hand and support the animal's chest or forefeet with the other (keeping them in essentially the same position as it is crouching in to begin with) you can probably move them safely to the more preferred substrate. But the success of that is all in the timing with what is happening at the moment. Sometimes this kind of quick move just surprises and frightens the cat so it isn't a good option.

We found that it was probably kinder (and usually faster) to just get something in front of the cat to catch the barf. Having a few sheets of newspaper handy in those locations to just grab and slide in front of them can work very well. Then cleanup can be just folding the paper up. Sometimes kitty will run or move when you try this so you have to be quick and easy about it, even timing it so that you get it there when they are just about to eject so they don't run.

Next, of course, is trying to figure out what is causing the need to barf. Hair ingestion during washing (hairballs) is a big one and getting your cat used to being combed or brushed with an appropriate instrument (visit your local pet or cat emporium to find the right item) during those times when they are shedding a lot of underfur is the key. It has to be an enjoyable experience for them (and you), so take your time in developing that as a frequent routine. There are also remedies that help the fur to be passed through the gut without needing to be ejected by mouth, tasty paste that a cat will eat that catches the hair and helps it pass through. I defer to your vet for the best advice there.

We discovered that our cat was horking down whole kibbles of dry food without breaking them up or chewing them at all. The advice we got from our vet was to put some large round stones (clean river rock larger than food size) in the food bowl. The cat had to pick up individual kibbles to eat and thus slowed down in eating and as a result the food was better masticated and less barfing occurred.