I've flown my kites on a variety of hills in the Peak District, Lake District and Scotland. Usually strap one on the rucsac if I'm out on my own. It adds an extra dimension to a good day on the hill.
I find the beauty of flying in the hills (in addition to the spectacular situations) is the way in which you can use the shape of the hill to create more or less wind depending on your needs. Turbulance depends on the shape of the hill. I try to pick a spot such as a col or a shoulder or summit that is likely to give less turbulant air. Choice depends on the general layout of the land and current wind direction. General rule I work to is that wind speed will increase as it runs on to a slope and decrease on the lee side. I do not tend to fly over edges due to the likely turbulence and the increased risk that the kite will be dashed into the rocks. Also you will have less scope for recovery and limited room for manoeuvre.
I try to take a choice of lines (short 15 metre to 25 metre) to allow for the limitations of the ground. Choice of kite depends on the likely wind as always but I would recommend one that is reasonably robust as you are likely to encounter rocks as well as rough ground. Ironically rough ground will often cushion a landing but can make it more difficult to take off again. Snags with the lines are less with shorter line but you have less space to fly. I tend to take my stronger lines after I broke a set on the summit of one of the Ben Lawers group on an icy wintery day. Last thing you want to be doing is chasing a run away kite on dangerous ground. Too may people get killed chasing after a rucsac or glove that has slipped or blown away in the wind. Take a Karabiner to clip your straps to you rucsac as the ground will not always take a peg. You will not have the same scope for movement and it is worthwhile checking the ground underfoot for rocks, holes and bog. Rounded hills give better wind and you will often find sweet spots in the air if you hunt around.
Over the years I have flown a wide range of kites on the mountains including Outerspace UL, Gemini, Stranger and Erasor with only minor damage to the Gemini when flown in glacial outflow in Iceland.