  We extend existing results on locally nilpotent differential polynomial rings
to skew extensions of rings. We prove that if $\mathscr{G}=\{\sigma_t\}_{t\in
T}$ is a locally finite family of automorphisms of an algebra $R$,
$\mathscr{D}=\{\delta_t\}_{t\in T}$ is a family of skew derivations of $R$ such
that the prime radical $P$ of $R$ is strongly invariant under $\mathscr{D}$,
then the ideal $P\langle T,\mathscr{G},\mathscr{D}\rangle^*$ of $R\langle
T,\mathscr{G},\mathscr{D}\rangle$, generated by $P$, is locally nilpotent. We
then apply this result to algebras with locally nilpotent derivations. We prove
that any algebra $R$ over a field of characteristic $0$, having a surjective
locally nilpotent derivation $d$ with commutative kernel, and such that $R$ is
generated by $\ker d^2$, has a locally nilpotent Jacobson radical.
