In this paper we introduce a new optimal transport problem which involves roughly a finite system of simultaneous time-parametrized transport which favors merging paths for efficiency over various time intervals and involves continuously differentiable transitions at the mergings (as with train tracks). We will describe suitable spaces of parametrized networks, topologies, and functionals, and then give an existence and regularity theory. Along the way we obtain necessary and sufficient optimality conditions applicable at times of various mergings. Additionally we introduce the problem of finding minimal surfaces in S3. In particular, we are interested in whether a certain minimal Mobius band is the unique minimal nonorientable surface with boundary a great circle. As this problem is too hard to tackle directly, we studied a related problem in a different bilipschitz space, the boundary of the bi-cylinder D2*D2.