I was involved in a joint dig a while back, enthusiasm waned - not sure if the (very small amount of) equipment was taken out, moved elsewhere or left. Need to chase that one up.
I'm involved in an "active" dig in a notable system - it's not been touched for well over half a year! On the last visit I removed a few small items, no doubt the next time we get round to digging there a few more unused items will be removed. I'd be more than a little p'd off if a good Samaritan decided to remove any gear without first asking. I think it's obvious it's work in progress, gear will still be in good condition and information is in the public domain. Rotten/abandoned gear is a different matter and usually fairly obvious and with a quick bit of investigation (other cavers or on here for example) you might locate the owners and whether or not it's active, wanted or not.
A surface dig I was involved in, a very short "cave", currently has a tub residing in it - I know it's there and I know it needs removing at some point (the dig is effectively abandoned). This is not likely to be visited by anyone, unless you happen to having a look around the area, so it remains a handy place to "store" a tub. I don't fancy getting my clothes mucked up and it's not often I find myself in the area with caving gear on. It will be removed at some point in the future.....
On a lighter note; ten years or so back I was wandering around a fell (Casterton or Leck, can't recall) and looking at numerous shakeholes. In one I found an abandoned broad headed shovel, it had a bit of grass growing over it, broken shaft, looked a little worse for wear. I picked it up in my arms, offered words of encouragement and rescued it from its predicament.
At home the surgery began and using a combination of duct tape, spare carpet and cable ties (that's what I can see, can't recall further detail) I created a very short shaft shovel. It was used to good effect in the above mentioned surface/short cave dig and another nearby soft-sediment cave where we had a good relationship. It is still in reasonable condition and currently enjoying a happy retirement in my garden storage box and is occasionally used for scooping up leaves or other garden material. Reuse and recycle