Non-Cai Mounts

Does anyone know about how Cai's tamed mounts work (if at all) with other routes? Like if you can transfer the tamed animals in the stable after completing Cai P1, or if you recruit someone and give them a unique mount, will that carry over?

It makes it feel real shitty to try and use any cavs on other routes tbh, but it'd also feel kinda rough to be "forced" to recruit and give units a mount in Cai's route to be optimal if that's how it works.