For leaders climbing with novice followers (let's just call them students) there is something you need to know... students will often fail to clean gear. Even well placed gear. And sometimes they just unclip the rope from it and leave it in the rock. Without having any idea that they did it. Sounds unbelievable, but it happens way more than you might think.

Me after being handed a single carabiner: "What was this biner attached to?"
Student: "Nothing"
Me: "The carabiner was clipped to the rope and nothing else? Floating in space?"
Student: "Yeah"
I went back down to retrieve my #1 camalot. TWICE! My poor #1 has been abandoned by two different students. But they always get the carabiner.

So... CHECK YOUR GEAR AT EVERY BELAY. You have a system for racking your gear, so it should be fast to make sure you have all of it. Every pitch.

Two points about stuck gear: (1) Having the students work on the piece will result in it being MORE stuck almost every time. Tell the students that if they can't get it, to just leave it for you. (2) Letting another climber clean your gear and get it back to you is ALWAYS a bad idea. Not because they will steal it (though they might) but because it is the absolute lowest priority item for them. Remind me to tell you about the time that letting another party clean a stuck #4 caused poor Josh to run from the base of Frogland to the summit of Frogland with a fully loaded rescue pack (two ropes, rack, personal climbing gear, water, food). You will cry for Josh's quads. You might also cry for the random non-COE person who nearly walked off a 1000' cliff in the process. You think I make these stories up, but I'm not that creative.

Anyhow, I'll give you three possibilities for stuck gear. I think they cover every situation.

1) You plan to rappel the route, AND you don't need the gear for the remaining pitches, AND you are certain that nobody will touch your gear before you get to it.
Clean the gear on rappel.

2) You want to get the gear back immediately and you can get two bomber pieces above your belay anchor.
Set up to lead the next pitch like normal. Rack your gear, stack the ropes, etc. Climb until you have a bomber piece of gear on each rope (or two bomber pieces on one rope if you are leading on just one rope). Then lower. Fun fact about climbing with double rope technique: Lowering on double ropes will almost perfectly equalize across your two top pieces once you are below both of them. Lower to the stuck gear, clean it, and climb back up and straight into the next pitch. This will take very little extra time. You will be reclimbing a pitch you just climbed, but now you are on toprope and not placing gear. The whole process will add five minutes to your day. Note that if the pitch traverses you will have to place directionals on the way down, but it will be no different from the gear that you placed on the way up when you led the pitch... so no worries.

3) You can't get bomber gear above your belay anchor (for example, at the top of a climb when you plan to walk off).
The solution I like here is a bit trickier, but it works much better than fixing a line and having to ascend. Tie a knot in the brake strand of the student with stuck gear while they are still at the piece. This knot will help you determine the amount of rope needed to reach the stuck gear without having extra. When the students are at the anchor and you have re-racked all your gear, tie in to the end of the rope with the marker knot. Put that rope on belay using your autoblocking belay device so that the knot is just on the brake strand side of the device. This will act as a backup and makes sure you can reach the stuck gear. Show your students how to belay in autoblocking mode. If they have never done it before, they will be amazed how difficult it is to pull the rope through the device. Smile and tell them that it was hard for you too, pulling both ropes through for the previous nine pitches. Now rappel down the climber strand. You might need to borrow a student belay device since yours is on the rope in autoblocking mode. Clean the gear. Climb up, belayed by the student. Again, for a traversing pitch, place directionals on the way down.

Those are my thoughts on stuck gear. Obviously in an emergency you can leave stuck gear (or any gear) behind. But a student's inability to clean a piece of gear does not constitute an emergency. If it did, COE's brand new racks would already be down to a single cam, four nuts, and one oval carabiner :)