There are different types of UAVs "drones" for a more buzzword-y term, for different uses. The bigger boys such as the Predator, Global Hawk, etc use satellite navigation. There is most definitely a "lag" of sorts for those types of UAVs. The Shadow and the Hunter however, are line of sight. It limits their range severely (I got one out to 166km from the control station), but they are just as dangerous, albeit smaller and with less endurance for extended missions. It doesn't matter, they are cheap and you can launch lots of them :)

I've got 480 combat flight hours from Operation Iraqi Freedom II, from 2003-2004. We CONSTANTLY were targeting moving things and never had any real issues with it. You just use the right drone for the mission at hand. The military has dozens of different types of drones, but only the bigger ones make the news on a regular basis. The Shadow is being phased out for those types of missions in favor of the MQ-1C Gray Eagle.

Now for air/air stuff, they call those UCAS (lookup the X45-B / X47-B), the tech is still fledgling enough to need another 5-10 years before it will ever become a reality. On that, there simply isn't good enough software... yet.