We had a similar situation. A neighbor had piled up soil on his property and did not intercept it before it reached our property. We asked him to do that. He didn’t want to. We suggested: we do it, costs split fifty-fifty. No, he wouldn’t agree! But it’s fine like this now (no, it’s not fine, the soil kept sliding more and more onto our property with every rain shower, which we have tried to stop so far with driven-in steel rods and plates).
Legally, he should of course have intercepted it; we could have just left it like that and waited until everything slid onto our property, then applied to the building authority requesting that he intercept it because the material was on our land, waited for the authority’s response, possibly taken legal action – and the whole thing would probably have dragged on for years. Then we could have said “too bad for you!” and “you won’t do that over our property”. Which would have meant that he would have had to do it on his own land. Quite complicated, since larger machinery can’t get in there anymore. So it would have taken even longer.
We roughly estimated: 3 - 4 years until we would have enforced and gotten the wall that was his responsibility. In that time, our garden would always have been a construction site, constantly dirty from the neighbor. Sure, unlawfully, but that wouldn’t have made it nicer.
In the end, we built a wall at our own expense. If the neighbor wants anything from us now, he can go f*** himself.
It has already been suggested here: try to establish contact, maybe through the real estate agent. Point out that the enclosure was unlawfully on your property and that this is exactly what causes the described problem now, and try to find a joint solution. Maybe they aren’t all so awful and will cooperate. Not everyone is such an asshole like our neighbor (he is known and notorious throughout the whole village here and annoys everyone).
Otherwise: bear the costs yourself while looking for a cost-effective solution. I also believe that DIY work is definitely possible here. But even then, make sure to get permission to enter the property for that.
The alternative would be to wait until new owners arrive and then make it clear to them that the enclosure was unlawfully placed (namely on your land), etc. etc. – do you want to have a construction site that long? That would be too stupid for me.