I can only tell you what I know from the little the MDs told me when they diagnosed me with Hiatial Hernia about 17 years ago - this is a rupture of a portion of the Diaphragm near the area through which the Esophagus passes on it's route to the stomach.
Don't know if this has any bearing on the shoddy diagnoses I was given, but it happened when living in South Florida, was given a Barium X-ray, they gave me a cup of stuff to drink and all they said was "drink it", they gave me no warning the stuff was going to instantly make my chest rapidly expand and feel like I was going to choke from so much internal pressure (kinda like eating way too many candy Pop Rocks in one mouth full). Then they sat me down in front of this funky tv screen where I could actually see my innards in a strange, live, moving, black and white picture. Afterwards, they told me very little other than "yup, you have a hiatial hernia" (this is litereally what the nurse told me, the doctor who sat only a few feet away from me during the exam never spoke one word to me). The nurse took my check, patted me on the back, wished me good luck and gave me a very very very very long list of foods to avoid because they were suspected to aggravate my situation. I think it would have been much easier for them to give me a much much much much shorter list of foods suspected to NOT aggravate the situation.
My situation is also complicated by Esophagael Reflux - an erratic functioning of the Esophogael Sphincter (possibly related to the Longitudinal Muscle you mention), that bottom part of the Esophagus (kind of like an anus) that opens to let stuff into the stomach, and to allow air/gas to escape back up the Esophagus (belching) but otherwise prevent acid and solid stuffs from backing up through this opening. Mine doesn't always operate this way, frequently stuff other than gas/air does back up (nasty heart burn), and frequently a portion of the stomach migrates into and through the rupture in the diaphragm (causes chest pains like heart attack).
After many years, I've learned to live with and adapt to this situation, but this from a mainly mainstream medical approach. It's only been in the last year though that I stumbled upon the MH forum, so I am hoping that as I gradually get better at obeying nature, this problem will further lessen.