  We introduce a new variant of the weak optimal transport problem where mass
is distributed from one space to the other through unnormalized kernels. We
give sufficient conditions for primal attainment and prove a dual formula for
this transport problem. We also obtain dual attainment conditions for some
specific cost functions. As a byproduct we obtain a transport characterization
of the stochastic order defined by convex positively 1-homogenous functions, in
the spirit of Strassen theorem for convex domination.
