CWD is the reason AZA deer populations have fundamentally fallen apart. It's more or less impossible to import deer from other countries except Canada, who also ban the import of deer from other countries except America. I believe small neotropical deer can squeeze through easier than Old World or large cervids, but even this involves so many hurdles it seldom if ever happens. Many state in the US flat out ban the importation of deer from other states as well, causing a lot of zoos to abandon deer programs. IIRC Louisiana changed their CWD regulations to now ban the importation of deer, essentially stranding Audubon's Barasingha herd from the rest of the fractured zoo population. New York and California both have regulations which allow the movement of deer in and out of the state as long as they are tested before departure and upon arrival. It's still a hassle and expensive but it's a lot more sensible than what most states do, and this is why Bronx and San Diego have become the two major zoos sustaining large Asian deer populations.
CWD regulations for most states are lessened for certain native species (which is bizarre given it's the native species most at risk) and for smaller species, which is why pudu, muntjac, American Elk, and to a lesser degree Tufted Deer have managed to spread out more (in addition to most of them being smaller and therefore easier to house).
I remember reading some time ago that a farm here in CT wanted to import some Domestic Reindeer and they ended up engulfed in a multi-year legal battle with the State before the regulation was amended to say that certified individuals can import Domestic Reindeer, but only for the holiday season and then they must leave again...
~Thylo