For ${\cal X}$ a stratification with suitable regularity, in particular for any Whitney stratification and, via regular embedding, for any abstract stratified set, time-dependent vector fields are used to prove an extension theorem for diffeomorphisms near the identity defined on strata of a given dimension. Then it is shown that after isotopy a stratified map $h\,{:}\,{\cal Z} \,{\longrightarrow}\, {\cal X}$ can be made transverse to a fixed stratified map $g\,{:}\, {\cal Y} \,{\longrightarrow}\, {\cal X}$.