Convergence of sequences of inverse limits

We consider the following question. Let X   be a compact metric space, and let (fn:X→2X)n=1∞ be a sequence of upper semi-continuous set-valued functions whose graphs converge to the graph of a function f:X→2Xf:X→2X in the hyperspace 2X×X2X×X. Under what additional assumptions does it follow that the corresponding sequence of inverse limits (lim←fn)n=1∞ converges to lim←f in the hyperspace 2∏i=1∞X?We give two nonequivalent conditions which generalize previous answers given by Banič, Črepnjak, Merhar, and Milutinović in 2010 and 2011.