A quarter mile is typical for European bees. Sometimes shorter and sometimes further if a suitable location is needed. Seeing a swarm for the first time is an amazing thing. If coming out of a hive, you hear a loud roaring sound. And then bees pile out by the thousands and get airborne. You can walk in the middle and your are engulfed in he middle of the bees flying. They WILL NOT sting at this point and are just looking for a place to settle. They usually settle fairly nearby once, for a few hours or overnight, if you are lucky. Then when the scouts clear the way, they become airborne again and stop when they reach their new home. I said if you are lucky because hopefully they are where you can shake them into a new box and keep them around vs losing half your bees in a hive. A hive may throw multiple swarms and this is not wanted, because the more bees in the hive the more potential honey. In my area if a hive swarms even once, the honey crop is done for that year.Oh yea, I have seen swarms flying, but usually at least near the tree tops till time to light. It is not unheard of to see a swarm at car level crossing a highway.
An AHB hive may fly a distance, stay a day or so and move again. Sometimes multiple times. This is why they spread so easily.