From what I know of the activity, it involves lots of squeezing yourself and your gear through small passages and there's often some water involved. Also, I would imagine, you wouldn't want to risk flying into walls (or people). So, if you need just a tool for exploring the walls and indentations of larger caverns, the best drone would be very compact in transportation, waterproof, stable and have a collision-avoidance system. A controllable camera gimbal is a big bonus for looking in different directions while hovering in place. That feature set suggests something in the Mavic line of DJI drones since they have all of it. They are compact, foldable, rugged, waterproof, stabilized, gimbal-equipped and collision-avoiding. If you want a recon drone, that would be it.
That said, while DJI drones can be flown indoors (and even have a special "tripod mode" that limits their forward speed), there seem to be some quirks with the position-stabilized modes there. Optical stabilization can fail in darker environments, the drone may wander around erratically for some seconds if it acquires some satellites through a window, for example. If you keep to the attitude mode, however, you should be fine.
All in all, if all you need is a camera platform/reconnaissance drone, get a Mavic (one of the smaller ones probably) and mount some powerful LED lights on it. Also, a good idea may be to get some prop guards with it despite the collision avoidance just to be extra safe. You don't want to lose that many dollars (or cut your buddy with a prop) in a cave.
Now, if you want to have fun with the drone, that's another deal entirely.
Large caverns have been flown by freestyle and race pilots on several occasions with great success. Depending on the size of the cavern you expect to fly, you might want anything from a larger (75mm+) brushless micro quad to a 4-5 inch quadcopter for huge spaces. The most universal is probably something like an 85mm whoop. Prop guards are a good idea in any case, ducts are probably not if you're looking for performance. Conformal coating all electronics to protect them from moisture is a must. Still, be prepared for the possibility of losing the drone. Also, acrobatic drones are not very useful for looking over stuff in a hover due to the fixed camera uptilt.
A digital video system is good for rejecting multipath signals in enclosed spaces, but note that all current digital systems are way bulkier than analog ones, so I'd probably go with a good analog transmitter and Rapidfire or similar.