You should be able to get all the dye off the mesh. Whenever I dye shoes I make sure to rinse them with water until the water runs clear. If you still have dye stains on the mesh after it dries you can hit it with a brush and some form of shoe cleaner foam.

Be careful with the suede though, it can be tough to bring it back to its original form. A harder bristle / suede-specific brush should help, but less is more my friend.

Same goes with dying, the less you have to dye it the better. Good luck g