I have a question very similar to one asked by wxfox4god on 12th June 2013 ("Arcmap 10 restrict Flow Accumulation") that was answered by whuber. (I have had to post it as a new question as I do not have the reputation to post a comment.)
The original question was about performing a version of the ArcGIS flow accumulation tool that is restricted to only accumulating cells from within a given distance. So for every cell in the raster the process finds how many cells within a certain distance would flow into that cell. whuber suggested a clever workaround using focal statistics iteratively.
However, my problem differs in that I need to find the weighted flow accumulation for every cell in the raster based on cells only within a certain distance. So for every cell in the raster I need to know the sum of all of the values from the cells that flow into that raster from within a specified distance.