"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.
Texas Penal Code Chapter 9 Sub Chapter D PROTECTION OF PROPERTY
§ 9.41. PROTECTION OF ONE'S OWN PROPERTY.
(a) A person in lawful possession of land or tangible, movable property is justified in using force against another when and to the degree the actor reasonably believes the force is immediately necessary to prevent or terminate the other's trespass on the land or unlawful interference with the property.
(b) A person unlawfully dispossessed of land or tangible, movable property by another is justified in using force against the other when and to the degree the actor reasonably believes the force is immediately necessary to reenter the land or recover the property if the actor uses the force immediately or in fresh pursuit after the dispossession and:
(1) the actor reasonably believes the other had no claim of right when he dispossessed the actor; or
(2) the other accomplished the dispossession by using force, threat, or fraud against the actor.
Texas has that too. At least no duty to retreat in ones own domicile. See subparagraph (b)
§ 9.32. DEADLY FORCE IN DEFENSE OF PERSON.
(a) A person is justified in using deadly force against another:
(1) if he would be justified in using force against the other under Section 9.31;
(2) if a reasonable person in the actor's situation would not have retreated; and
(3) when and to the degree he reasonably believes the deadly force is immediately necessary:
(A) to protect himself against the other's use or
attempted use of unlawful deadly force; or
(B) to prevent the other's imminent commission of aggravated kidnapping, murder, sexual assault, aggravated sexual assault, robbery, or aggravated robbery.
(b) The requirement imposed by Subsection (a)(2) does not apply to an actor who uses force against a person who is at the time of the use of force committing an offense of unlawful entry in the habitation of the actor.