Super Mario Flows in NP?
It's been a long time but I am pretty sure this problem is in P.

I wrote my PhD dissertation about this in 1995. See "Efficient Dynamic Network flow algorithms" by Bruce Hoppe submitted to Cornell CS dept. Online at http://dspace.library.cornell.edu/bitstream/1813/7181/1/95-1524.pdf

See Chapter 8 "Extensions" section 8.1 about "mortal edges"