It is entirely possible to do something like this and would be encouraged under most circumstances since it would allow for exchanges to have a more consistent and evened-out exchange rate, something good for outward appearances but likely bad for individuals who make speculating money off of the large differences between each exchange.

Challenges are transaction fees and transaction times; both can make you miss out on a deal. I have no idea if others do this kind of trading, but likely some people do. I've thought about it myself.