In Joshua 3:3 (NASB)
and they commanded the people, saying, “When you see the ark of the covenant of the Lord your God with the Levitical priests carrying it, then you shall set out from your place and go after it.
And in Numbers 4:15 (NASB)
When Aaron and his sons have finished covering the holy objects and all the furnishings of the sanctuary, when the camp is to set out, after that the sons of Kohath shall come to carry them by the poles, so that they will not touch the holy objects and die. These are the things in the tent of meeting that the sons of Kohath are to carry.
Isn't carrying the ark a function for non-priests?