Most woodworkers I know, including myself, walk along with the board when jointing, to some extent; but for long boards they stay in one place either at the infeed or outfeed side and use the push pads hand-over-hand to "walk" the board across. You should put downward pressure on the infeed table when starting the cut and on the outfeed table when ending the cut, but there are two schools of thought for what happens in the middle. Some keep pressure on the infeed table and others keep pressure on the outfeed table. Personally, I think it makes more sense to put down pressure on the outfeed table, but I have woodworker friends who do the opposite and produce much nicer work than anything I've ever made. As long as you stick to one or the other, you should get a good result (except perhaps at the very ends), even if your infeed and outfeed tables are ever-so-slightly out of parallel.
In terms of safety, you should try to keep your hands at least 6" away from the cutterhead at all times, which also means you shouldn't joint anything shorter than about 18-24". As Doresoom mentioned, you should use push pads. Because you're usually working with longer pieces, push sticks and push shoes might not work as well on a jointer as they might on a table saw, though I suspect Matthias Wandel might still use push sticks on a jointer.
Ideally you should never pass your hands over the cutterhead, even if you are using push pads, but this can be very awkward at first, and at least one of the more famous instructional Internet woodworkers walks along with his workpiece and passes his hands right over the cutterhead (with push pads, of course). You can use featherboards for face jointing, but most jointer fences are not tall enough to use featherboards for downward pressure when edge jointing wide boards.