  A search is performed for $D^{(*)+}_{sJ}$ mesons in the reactions $pp \to
D^{*+} K^0_{\rm S} X$ and $pp \to D^{*0} K^+ X$ using data collected at
centre-of-mass energies of 7 and 8 TeV with the LHCb detector. For the $D^{*+}
K^0_{\rm S}$ final state, the decays $D^{*+} \to D^0 \pi^+$ with $D^0 \to K^-
\pi^+$ and $D^0 \to K^- \pi^+ \pi^+ \pi^-$ are used. For $D^{*0} K^+$, the
decay $D^{*0} \to D^0 \pi^0$ with $D^0 \to K^- \pi^+$ is used. A prominent
$D_{s1}(2536)^+$ signal is observed in both $D^{*+} K^0_{\rm S}$ and $D^{*0}
K^+$ final states. The resonances $D^*_{s1}(2700)^+$ and $D^*_{s3}(2860)^+$ are
also observed, yielding information on their properties, including spin-parity
assignments. The decay $D^*_{s2}(2573)^+ \to D^{*+} K^0_{\rm S}$ is observed
for the first time, at a significance of 6.9 $\sigma$, and its branching
fraction relative to the $D^*_{s2}(2573)^+ \to D^+ K^0_{\rm S}$ decay mode is
measured.
