Somewhere right near your link for steering is his examples for obstacle avoidance -- be it for fixed objects or dynamic ones.

Edit: Ugh... clicked over there and was reminded that his java apps don't exist anymore and so one of the best websites for n00bs looking to get into this is trashed. Sigh.

If you already use path following to "drive" you can avoid collisions with static objects by simply adjusting the path your drones follow until a drone can complete the whole race without crashing. 

You'll probably need more data than a 3D curve representing the optimal path, such as

• velocity limits by location and facing
• next waypoint to aim for, by location and facing

Then you only need to worry about collision with other drones; your drone must override the normal path following to slow down and avoid collisions.

It is relatively easy to compute, or to estimate conservatively, the set of objects the drone is going to collide with before a certain time increment; if it is nonempty the drone can simply brake, and it will fall behind if the obstacles are other racers. Regarding braking, since a drone doesn't have to worry about losing grip like a car, you could apply maximum thrust in the backwards direction for the shortest time necessary.

There's tons of other information out there on "steering" and "obstacle avoidance" and whatnot.