Do mirrorless (hybrid) cameras like the Sony NEX-x or the Panasonic Lumix GFx let you set manually the diaphragm?
Yes. Every single mirrorless ever made offers full-manual controls. Some just hide them better than others! Having a mode-dial also helps to quickly set the camera into Aperture priority mode. Fuji lenses for mirrorless cameras all have an aperture ring right on the lens which makes things very easy.
If you are keen on using semi-automatic (A/S/T) or manual (M) modes, I suggest you look for a camera with at least one control-dial. Two is better if you use Manual mode as there is nothing to press to toggle between both parameters. Some mirrorless cameras even have three control-dials.