Those are the two most pernicious brush types to remove/control/eradicate that there is. See your County Agent... get his recommendation... much depends on how fast you need relief, how much you wish to spend, how dense the infestation is, how large the plants are, etc...
I can't tell where you are, but I suspect somewhere in south Texas...
Learn about Reclaim.... this works . I've almost completely eradicated an infestation of 3 to 18 ft high mesquite on 200plus acres... took years to do and I still have beans sprouting... you will too...
If you use mechanical control.. unless you PULL all the roots, they will come back from the roots.
If you have larger mesquite, you can use it for BBQ and recover some of the removal expense. If you have VERY large mesquite... 2 feet in radius or larger, it is furniture grade and people will PAY a lot to harvest it.
Suggest you do a detailed google type search " mesquite brush removal texas" . Texas A&M has some good articles/info on the net about demonstration projects.
The beans continue to sprout as long as 15 years...so, be prepared for an annual battle.