Thank you very much for you answer Mark, it has been very helpful. I understand more what I'm dealing with now, and I see how to move forward; since we keep indices only for a certain amount of time, we'll try to drastically reduce the number of indexed fields in the new indices, so that when the old ones rotate the problem goes away.
In the meantime, I was told about frozen indices, which could help in the short term: we have to keep data for a certain amount of time, but we usually don't use the oldest part, so the decrease in search performance shouldn't be too big a problem.
Thanks again!