A super fast/easy check is to create a bounding sphere that encapsulates the move, from the desired_pos - current_pos vector, then just do a radius^2 check between the two objects you are doing a collision detection for. The move between frames will be small, and this test actually tends to require fewer calculations than traversing a bsp/quad/octree to find the leaf that something resides in. After that, if you choose, you can dot these two vectors together, because there are cases where they may be inside this imaginary bounding sphere but the objects are moving away from each other (just test the sign of the dotproduct). No normalizing, don't need to traverse a partitioning tree.
I needed a method to easily reduce the working set of testable objects.