There are two types of movement, expansion into unoccupied territory and movement within an established range.
I think that some deer will quickly expand into suitable unoccupied territory adjacent to their current range, roe and muntjac in particular have spread into new areas at a phenomenal rate over the past 50 years. However if they are in occupied ground I think the amount of movement reduces to something more natural and only the young bucks move significant distances when pushed out by territorial bucks. If you see a roe buck in a particular location the odds are that he'll be there at the same time the following day. Muntjac also tend to stick to an area.
The "herd" deer such as sika, reds and fallow have different dynamics and the population is usually centred around the location of the female herd as the males might go further afield but will come back to the females for the rut. The female herds are usually fairly loyal to their general home range, which can be quite large, but I think that pressure from stalkers or poachers might prompt a wholesale move and fallow are notorious for being erratic in their pattern of movement and reaction to pressure.
In my local area of North Wales the fallow move about a fair bit within a large but generally well defined range, you might see them in a field today and not again for a week or two, but they do not often move into completely new areas unless very pressured. There has been an increase in the local cull over the past few years, woods that were previously sanctuaries were culled and it may be coincidence but, at the same time, fallow started to appear just outside their normal range limits where they had rarely if ever been seen before.
The rate of territorial expansion as opposed to general movement of a species is also clearly going to be higher when none of that species are present in adjacent suitable areas any increase in population results in territorial expansion and the rate of expansion of reds in East Anglia is a testament to that.
i know least about sika but it seems that they might be more demanding on what constitutes suitable territory, less likely to move away from their home range and less likely to move long distances unless pressured, however within their defined range they can be unpredictable and seeing them one day does not mean they'll be there the next day.