Mark all of the mesquite trees that you want to remove with spray paint. Make sure that the marks are clearly visible from all sides of the tree. Marking is especially important if there are other trees in the area that you want to keep and if someone else will be operating the dozer.
Use a dozer that is at least a D6 and more than 80 horsepower. Make sure you have a front mounted blade plow. Approach the mesquite tree from the opposite side from which you want it to fall. A few feet from the tree, lower the blade into the soil slightly and advance forward. You want the blade to sever the main part of the trunk off of its roots about a foot underground.
Push the mesquite tree down. As it begins to fall you can raise the blade up on the trunk for more leverage.
Use a chain to drag the tree out of the area. If you knock down several trees you can combine them into a pile to clear the area.