It does sound a little like reflux. If there isn't anything being brought back up, it may be silent reflux, which is why the baby is more settled when they are upright, as without gravity, irritants can back flow up into the osephagus and this acidic content hurts!
There can be an issue with the sphincter muscle between stomach and the osephagus which may require further investigation however this can be invasive for the infant.
Things you can try is smaller feeds (if this doesn't irritate bub), avoid things in your diet like spicy foods, citrus fruits, tomatoes and other acidic food, fatty foods and caffeine which may increase likely hood of reflux.
During the birthing process, particuarly if it has been a difficult birth, there can be compression of structures which supply through the stomach and diaphragm which may also effect function. Osteopathy can elevate this anduse gentle treatment to address these issues which may help with the childs symptoms.