Let $(R,m)\rightarrow (R',m')$ be a local ring map (of noetherian rings) which is étale local, namely it essentially of finite type, it is flat, $mR'=m'$ and the field extension $R/m\subset R'/m'$ is finite and separable.
Which properties on $R$ ensure that $R'$ is equidimensional? For example, if $R$ is a domain, is it true that $R'$ is equidimensional?