Back in my backpacking days, and later motorcycle camping, I used a ground cloth for the dirt/leaf/pine-needle protection, then would use a towel of some sort to dry it off as I rolled it up. Fold, wipe, flop, wipe, roll half a turn, wipe it off, roll half a turn, wipe it off, repeat until done. This would take care of the bulk of the dirt and water or moisture issues. Then as noted above, later in the day roll it out to let it air out some while having a bite to eat or looking at maps or whatnot. If it is raining day and night then just let the moisture freshen up when you set it up each night to hopefully avoid the rot and mildew issues.

That's what worked for me in the past, adapt and adjust to suit your own preferences and needs.