Animals can despawn, as PALU said. I don't know the exact mechanics involved, but I would assume animals don't despawn from the current wilderness map tile. Nearby and/or recently-visited tiles (visited on the zoomed-in map) are probably reasonably safe as well. It's also possible for animals to spawn on islands, which setting traps allegedly influences (again, I don't know the exact mechanics, but it's beneficial so long as you don't upset the forest spirits).
Animals can't cross water if it means entering the water. I don't know about crossing water where the land is connected diagonally, but I would assume so. I've seen them pass through diagonal gaps in fences. If you're worried about animals escaping, just block their path with a trap.
Predation does exist in URW, so you could lose your prey to a predator, unlikely as it is. But, hey, predators are more valuable anyway!