Chop several cuts into the mesquite tree trunk with an ax.
Wearing rubbers gloves, combine 3 parts dish detergent with 1 part tree herbicide in a bucket. Mix thoroughly with a paint stirrer.
Pour the mixed contents into a spray bottle and liberally coat the trunk where the cuts were made. Allow the herbicide/detergent mix to soak in and work as long as suggested by the manufacturer. This will seep deeper into the trunk and begin to kill it from the inside out.
Cover the trunk in a tarp that's secured with rope. This will repel sunlight and rainwater, starving and killing the trunk from the outside in. Check back in four to six weeks, the trunk should be dead and start to decay.
Uproot the trunk using a winch or machine excavator once it has died.