As with most such issues, that depends on the circumstances and the state in which they occur. Texas allows for use of deadly force to protect and/or recover property, under most circumstances. In such a case if the bad guy is making off with your property, say your vehicle or your horse, you may chase and use deadly force during the pursuit.
"In such a case if the bad guy is making off with your property, say your vehicle or your horse, you may chase and use deadly force during the pursuit."
That's pretty much unique to Texas, then. Even states with "castle doctrines" would not support that.