We have been successfully cleaning dressers and Hunter Douglas cloth blinds for many years.
@Mikey P I don't get your showering with clothes on analogy. Thinking that way, you could apply the same logic to cleaning carpet or upholstery, which you do onsite using the same tools we use for cleaning drapery and blinds
We find most can be wet cleaned with no issues. But we do dry clean the wine that require that. We have the US products dry cleaning machine. We used to wet clean with that as well, but found usinfrhe teuck mount and upholstery took to be more efficient. We've never had an issue with them falling apart, delaminating etc. I don't see the reason to be skeptical or shy away from it.
As a matter of fact I had a woman call yesterday from an hour and a half away because she can't find anyone to clean her cloth blinds and she heard about us. I sourced it to another cleaner since it's too far for us for a small job and she didn't want to pay travel charge or wait till a slower time for us.