Escalation is the act of moving conflicts to a different arena.
Usually this means that a conflict begins with less lethal means, and, as people become more willing to kill and die over what's at stake, advances to more lethal means.
But, regardless, when you Escalate, all the Fallout that goes out to anyone who Takes a Blow from you comes from your new arena, and the Traits, Relationships, and Belongings you can call on are limited to the ones appropriate to your new arena.
...a different, appropriate, arena.
Most conflicts can be reasonably resolved in several different ways; if you're trying to get Mr. Eastman to account for himself before the flock, you could talk him into it, sure. He could also show up there hogtied or at gunpoint.
However, this isn't true of all conflicts. If you're tracking the Dulles Gang through the desert to their hideout, that's notionally a non-physical conflict (acuity + heart) to follow their tracks which might escalate to physical (body + heart) to endure the desert, but that's about as high as it gets. You can't pull your Bowie knife and start hacking at the cactuses or draw your gun and vent your frustration with three rounds right at the uncaring sun. Well, I mean, you can, but none of those things actually help you track the Dulles Gang, so it's not appropriate to escalate the conflict to weapons or guns.
Similarly, with something that's explicitly an arm-wrestling match or a quickdraw conflict, that's a space that's purely physical or purely guns as determined by the original stakes of the conflict - who draws first, who's better at arm wrestling. If you move to a different arena you've stopped gunslinging or arm-wresting, which is the only way to get the stakes.
However, escalating isn't necessarily the only way to change the character of what you do in a conflict. If, for instance, your Dog has learned some bad lessons on the trail and grabbed a Trait of "I'm a no-good snake with a mouth full of poison 2d6+2d4", somebody like that would have no trouble cussing somebody else out to throw them off their game, without necessarily leaving that narrow arena.