I want to say there's a canonical variant on a dropship specifically meant for shipping cattle and other agricultural critters. Maybe of the Princess class?
I agree with the previous posters that avoiding the task of moving full-grown live animals would be the first thought; but since you cannot always avoid it, there ought to be some sort of cattle transport DropShip. I don't recall one mentioned in canon, but I have two designs statted and fluffed should CGL decide to take me up on my offer for another aerospace (X)TRO (working title: TRO:3157 :) ).
One is a large, Star League-era cattle transport built for colonizing purposes. It is conceivable in a universe and time where individual planets were hyper-specialized manufacturing centers and interstellar travel was cheap and plentiful. It is a variant of a large passenger transport for colonists, reconfigured for livestock.
The other design is a much smaller vessel specialized in animal transportation, with large inclosures, extensive medical facilities, all the bells and whistles. This is the kind of ship used when large animals of large value have to be shipped around - the Turin horse breeders would use such ships for their Liao Stallions, the Branthkeepers for Branth, and the Ghost Bears for, well, Ghost Bears. All of these animals have been mentioned to be shipped around on DropShips in canon. (Or on a somewhat more mundane level, Tabiranth or Galisteo Range-bulls.)
I imagine there isn't much difference between life support for humans and life support for animals, so construction rules wise such a lifestock transport DropShip would follow the rules for a passenger DropShip, perhaps with a markup on space/tonnage relative to cargo carried (i.e. the life support and living space for luxury quarters for 10 people would only support 500kg worth of livestock).
Zero-g isn't much of an issue as the DropShip will be operating under thrust most of the time. Handoffs on a pre-arranged bus line, booked months in advance, are the most likely approach. We've seen this done in canon for passenger vessels with the
Silver Eagle; switching JumpShips seems to be a fairly common, if perhaps expensive, thing.