I do not know how TO do it, but I do know a few DON'Ts...

The plants will not survive in the trailer unless the trailer is air
conditioned or open to air movement - when live plants are transported
by road, they are either on flatbeds or in reefers (trucker talk for
"refridgerated unit") The heat from the sun beating down on an
enclosed space will heat them untill they are almost cooked.

Provide the cats with their own litter setup that they have previously
identified as the proper place to go, otherwise your plants will become
fair game...

I suggest a little research to make sure the plants are not toxic to
either cats or dogs... they will be stressed by being cooped up, and
will want something to bat at or chew.... there are enough houseplants
that are bad for kitties and puppies that this will be a
consideration...

Another concern that has nothing to do with the plants... you may want
to consider a kitty carrier - cats are more territorial than not, and
if you have them loose in the vehicle, there is a rather high
likelihood that they will try to (and succeed in) escape when you open
the door, in an attempt to get back to known territory... I learned
this the hard way

good luck moving

Tony