Egg-dropping is a difficult and controversial technique. It makes all robos open to attack by the Umbrella Pod, even if they normally are not affected. An "egg-drop" can be performed with a robo that has considerable aerial qualities, an Umbrella Pod, and an Acrobat Bomb.
It involves exploitation of the "physics engine". From huge heights, the pods fall very fast, and can't slow down fast enough to stay airborne, causing them to fall to the height of even the shortest robos.
You must fire the Acrobat Bomb while you are on the ground, then you may air-jump to get good position on the enemy, then release the three Umbrella Pods. If done correctly, the pods should drop like rocks and dip below their normal position--hitting the target. If they do not hit, they return to their normal position and detonate. The most difficult part of egg-dropping involves getting to just the right angle so that your enemy cannot avoid the pods. To hit a strafing ground opponent, fire the pod in the direction that they are approaching. It will make hitting rhough lag much easier.
Stay beneath the enemy robo, and wait for it to land. You can also jump straight up while they're firing their gun and hit them with your own. Using the Gravity Gun or the Needle Gun can prevent takeoff.