  Skorokhod's M1 topology is defined for c\`adl\`ag paths taking values in the
space of tempered distributions (more generally, in the dual of a countably
Hilbertian nuclear space). Compactness and tightness characterisations are
derived which allow us to study a collection of stochastic processes through
their projections on the familiar space of real-valued c\`adl\`ag processes. It
is shown how this topological space can be used in analysing the convergence of
empirical process approximations to distribution-valued evolution equations
with Dirichlet boundary conditions.
