  For every compact surface $S$ of finite type (possibly with boundary
components but without punctures), we show that when $n$ is sufficiently large
there is no lift $\sigma$ of the surface braid group $B_n(S)$ to
$\operatorname{Diff}(S,n)$, the group of $C^1$ diffeomorphisms preserving $n$
marked points and restricting to the identity on the boundary. Our methods are
applied to give a new proof of Morita's non-lifting theorem in the best
possible range. These techniques extend to the more general setting of spaces
of codimension-$2$ embeddings, and we obtain corresponding results for
spherical motion groups, including the string motion group.
