We use shavings and have mats in both of our horse trailers. I do put extra shavings in the trailer when we go on long hauls (6hr +) to absorb the urine. We have never had any issues and I have gone in the back of the trailer when it was moving. The windows were open as well as the vents, and no shavings were moving around.
We did have a door at the side of the trailer open once (the top door of a 2 + 1 trailer) and the shaving were blowing around quite a bit. We stopped and closed the door but opened the window and it stoped the wind tunnel. The back doors can be opened with no issues though.
I know lots of people who trailer with shavings and a few who don’t. I think its really up to you.