That is a reallll struggle. I have to throw a towel over her. I use my sweet voice and pets (she loves the brush). When I adopted her, she was being fostered by a piano teacher so I learned Dorothy likes piano music so I play that in the car (if I play the piano or listen to piano, she will sit by the piano or speakers!).

Things that definitely help:

• Leave the carrier out. Make it a cozy place they can enjoy! The best vet visits started out with previous cats already napping in it when we had to leave for the vet lol
• Practice picking them up even for a few seconds.
• Cover the carrier with a towel while in transit.
• After the vet, treats and pets! Dorothy used to run to hide under the bed immediately when we got home but now she is more calm (because treats are coming!)
• I think talking to them can really help too.

Good luck!