hmmm, a tricky one. Shadow, I would say is perfectly happy to get into anyone's car and go off with them (though he tends to look alarmed when he realises I am not coming). The cats ... If they have something that smells of you, they will probably settle better if they do not see you in their new place. If you are there, and then you are not there, it will confuse them and they will look for you. If they are taken there, and they have home comforts, then they will probably settle quicker ... But that is my cats, and I know them and how they settled in here.
I do, however, try to be upbeat about it for them. Not weep wail and gnash my teeth (I do that after they are gone). Prolonged goodbyes are for the benefit of the human, not the animal and tend to distress them as they are distressed whenever we are upset ...