If elevated hydrogen sulfide is helping cause both gastroparesis and migraines, then there should some connection between the two. It turns out there is an association :
Gastroparesis is a chronic disorder manifested by delayed emptying of the stomach. Gastroparesis is a relatively common complication of diabetes. In a population of patients with symptoms of diabetic gastroparesis, the patients with cyclic symptom patterns had a higher incidence of migraine headaches (47 vs. 20%, p = 0.02) compared to patients without cyclic vomiting pattern (20). Migraine attacks are associated with delayed gastric emptying (21). This migraine-associated gastroparesis is a problem for the treatment of the migraine with oral medicines, like oral triptans (22). Initially, delayed gastric emptying was found during migraine attacks, now there are also indications that in the interictal periods migraine patients have delayed gastric emptying. However, the studies done so far have been small and inconsistent in their results (23, 24), so further research in this topic is warranted.
Could an H2S-synthesis inhibitor help prevent gastroparesis? If so, I first nominate the lowly common vetch. Obviously not for use during pregnancy (http://petticoatgovernment.blogspot.com/2016/09/common-vetch-and-phocomelia.html), but we know vetch contains beta-cyanoalanine, which inhibits two H2S-synthesizing enzymes (https://www.ncbi.nlm.nih.gov/pubmed/23488457). A second possible suppressor of H2S biosynthesis is L-aminoethoxyvinylglycine (AVG) (https://www.ncbi.nlm.nih.gov/pubmed/23488457), which is widely used to delay ripening of plums, apples, pears, and other fruits (http://link.springer.com/article/10.1007/s10725-008-9312-5, http://scialert.net/fulltext/?doi=ajcs.2014.320.333). AVG is even approved for use on organic fruit (http://www.sumitomo-chem.com.au/sites/default/files/literature/retain_progibb_cherry_technote_online_0.pdf). Gastroparesis patients are unlikely to get much AVG in their diet, seeing as fruit peels are warned against in many dietary guidelines for them; even if they did eat fruit peels with AVG sprayed on them, any resulting H2S-reduction might be trivial.
I'd love for someone to test whether common vetch or AVG can lessen gastroparesis symptoms. And perhaps migraines, too.