Sorry Bencho I missed your question before. I never clean the tubes. When I did case repair I would often take the foam out of the bottom and they really aren't that dirty. When you put your cue into the case the fabric is not touching the top of the tip so the only place where chalk accumulates is on the foam rubber at the bottom of the tube.
I have always looked it as if I had a cube of chalk at the bottom of the case. I never wipe the chalk off my tip when I put it away because once I have it chalked like I want it I don't want to mess that up. Of course I do things differently than a lot of folks it seems.
If you really want to clean the bottom there are a few ways. Before you do anything you should look inside the tube and understand what you need to do. I use the little super bright l.e.d. flashlights to inspect tubes. With those you can see all the way to the bottom of most tubes clearly.
1. Carefully take the nails out of the bottom and remove the bottom cap Before taking out the endcap make a mark so that you know which direction is correct when you pu it back. Then using a razor make x slices in the tape at the bottom that holds the foam rubber in. Then use your cue to push the foam rubber out and clean. Replace the foam rubber, put a piece of tape over the end, replace the end cap and the nails.
2. You can do the aforementioned ideas about putting wet wipes and the like on the end of a stick but you need to be very careful of two things - one that you can pull out whatever you put in - what you think is secure before you stick it in the tube often works it's way loose far easier than you think. - and two that whatever you use won't leave a bunch of fibers in your tube as paper based cleaning cloths tend to do. I would use something like a terry cloth towel and just make it lightly damp.
3. to get general dust out you can take rubber bands and a plastic bag and use the rubber bands to seal the bag to the top of the case and seal the other side to a vaccuum tube and let it suck the dust out. This won't clean the foam rubber at the bottom.
4. You can buy a case where the tubes are easily removable by pulling on them from the top. Some market this as a feature.

Just kidding. It's not a feature but is instead a defect in design (unless you like to clean your tubes a lot).
5. You could just put something in the tubes to cover up the dirty layer - I'd suggest a small piece of squishy foam rubber. This won't take up any room - a mm at most and will be a nice clean surface for your shafts.
That's about all the advice I have on this. Hope it helps you.