I don't think KB lifts in general have a higher mobility demand than barbell lifts or many bodyweight skills.
IMO, there are no separate prerequisites for thoracic mobility in order to do any KB drill, other than being able to do the drill with good form, with minimal restriction, and without causing pain or injury. In other words, you don't have to pass any separate test to qualify to do a drill, other than being able to execute the drill itself.
And there are a lot of workarounds to various restrictions. For instance, for military presses, you might be able to do single presses just fine, but double presses (which have a higher mobility demand) might be problematic (for instance, you might have trouble achieving a good lockout position, need to hyperextend the low back at lockout, or just have shoulder discomfort). With KB front squats, you might have to limit your depth. In my experience and observations, the catch position in the jerk (partially squatting under the bells as you lock them out) is one that is challenging for a lot of people, who either hyperextend the low back or are limited to a very shallow second dip). But those same people may be able to double MP or push press just fine.
That doesn't mean that you shouldn't address restrictions where and when they occur, but that they don't have to prevent you from training.
As far as specific advice on increasing thoracic mobility, IMO it's very individual. One person's pet drill will do nothing for many others. So you have to try out a lot of stuff and find what works for you by a lot of trial and error. That being said, here are some of my favorite high bang-for-the-buck shoulder girdle tune ups:
--Original Strength crawling and rocking variations.
--The David Allen band pull-apart super series:
--Mace and clubbell swinging.
--Dislocates using a PVC pipe (I didn't start noticing results from these until I did them very consistently and patiently over time, keeping my hands at a comfortable distance and not trying to force progress).
--KB snatches.
--The D I C K Hartzell band stretches (I have to write his first name that way or the language filter makes a mess of it -- the filter does not approve of the nickname for Richard), especially the set up with the band looped around the elbow and hand in the video below. I do my own variations of this kind of stretching, but I find that the band set up that captures the elbow makes a huge difference (and IMO should be widely used in physical therapy circles, although none of the many physical therapists I've worked with had ever seen it before):
--Various swings with light indian clubs (not necessarily the traditional indian club drills, but mainly ballistic swings in various patterns to explore the range of motion, some of them based on Scott Sonnon's Intu-Flow program, which is on YouTube).
--The Ido Portal diagonal stretch. This is one I've been practicing lately and it seems to unlock a lot of things for me:
Things that have been counterproductive:
--Any sort of wall slide variation.
--The bodyblade.
--Focusing on "packing the shoulders" when lifting overhead. In my opinion and experience, rigidly keeping the shoulders "down and back" (as "shoulder packing" is commonly taught and implemented) is a recipe for impingement.
Things that have been of marginal benefit:
--I, Y, T, W exercises.
--Scaption.
--Hanging from a bar.
--KB armbar variations.
Some general principles or strategies that I've found helpful:
--If it hurts (during or after), don't do it.
--Hammering away at trying to stretch the area that is restricted is not necessarily the best strategy. Often there's some other issue and you have to stumble upon some not obviously related drill that tricks your body into unlocking the restriction.
--Drills that are "low threat" and don't force awkward and uncomfortable positions. Train your nervous system to accept comfortable and non-threatening movement and gradually expand the range and/or load (for me, this means NO wall slide drills)
--Drills that get the muscles around the shoulder firing reflexively (such as the David Allen pull-apart series, OS crawling, clubbell and mace swinging).
--Drills that develop strength in the end ranges of movement (such as the Jump Stretch band stretches, and various other contract-relax stretches).
--Drills that move the joint quickly and ballistically through the range of motion, under minimal tension. (such as the light indian club swining mentioned above). I've found that the nervous system will accept moving ballistically under low tension through ranges of motion that are restricted and/or painful with slow motion -- there is less threat response and the nervous system is disinhibited from protectively tensing the muscles, restricting the range of motion and causing sensations of pain.
