Also check out 永代供養 (Eitaikuyō), which means entrusting the remains to the local temple in a communal burial plot/crypt. Typically you would pay a mortuary-service provider to close the family grave; remove the remains, clear the headstones, arrange for a priest to say the necessary rites. And then the remains are moved to the temple plot and interred there with more prayers and payments to the temple. It is a common/everyday service, and I think becoming more and more popular. If your relatives have a particular temple they are associated with, I'm sure the temple will help you out.
Your cousins will be glad to be rid of the burden of the gravesite. The gravesite caretakers are probably bugging them about upkeep and maybe even maintenance fees. Its good to be proactive about this. I forget how much the eitaikuyō costs. Its probably slightly more expensive than you imagine. Something like 1,000,000 yen, I think. (US$10,000 about).
Probably costs a bit less in Shikoku.