You can transport a lox easily on the largest boat. And I have lox and wolves around my main base in the meadows and stay close enough to have never despawned yet.
Even had a lox give birth whilst I was riding it miles away from base and the baby lox remained in that location long enough to mature and then be ridden back.
On the other side of the world I have two lox near a mistlands biome and I rarely visit it and they are still hanging around fine.