There has to be a standard way to transport livestock. I don't imagine they do well in weightless environments. I bet the animal freaks out. And if a horse loses bone density in zero-g the same way humans do, it'll be in danger of snapping its legs when returns planetside. My guess is that there won't be many interstellar racehorse competitions. A lot of that stuff is gonna be limited to a single system.
I'm also of the belief that there would be some kind of interstellar travel agency on most worlds. They'd have info on regular shipping routes, and would be able to save you money and time by setting up a mini-command circuit, or a series of them. Understand that these ships aren't waiting on you, you're just hitching a ride on vessels that happen to be in the same system at the same time. Just like an airline today, where you want to go from Santa Fe to New York City. You might have to go to Atlanta or Dallas, and then you've got a 3 hour layover before your next flight leaves.
Maybe you go to an interstellar tourist company a year before you make your trip, and they can tell you that in 8 months, there'll be a series of dropships where their schedules synch up. A jumpship will be leaving your system, going 30 light years to a neighboring system. Two days later, a different jumpship is scheduled to jump out, headed to star system #3. The next day, another jumpship is supposed to go from system #3 to system #4. From there, you'll layover in system #4 for three weeks. And after that you'll go to system #5, wait for four days, and catch another ship to your destination of system #6.
A service like that would be expensive, but nothing compared to the overall costs of travel. Who cares if you pay them 2,000 C-bills to plan your trip, if it's saving you hundreds of thousands. Travel would still be an affair of the rich, but you wouldn't have to set aside years to go from one end of the Inner Sphere to the other. Maybe on some of these dropships you're just sleeping in a converted utility closet, but you're not going to be there too long anyway.
Livestock are probably transported like this. While the dropship is accelerating at 1G, the animal doesn't know any different from normal. You've got a couple handlers there to calm them down during weightlessness (or some vets to sedate them), and you stick them in a padded room/tie them down so they don't hurt themselves. The livestock dropship has timed its journey so that multiple jumpships are synched up, and it travels 3 or 4 jumps before a layover on some planet somewhere. Then you wait for the next jumpship alignment, so your animals don't spend too much time weightless.
As long as you're willing to wait relatively long periods, it shouldn't cost too much. For many animals (like cows and things), if you want to introduce them to a new world, it may make sense to establish a herd on your layover planet with a local farmer, and then take the next generation of animals to your destination. Especially if there aren't good "connecting flights" and you're just going to have to wait for months.