  The decays of the $B^{+}$ meson to the final state $D^{*-}D^{+}_{s}\pi^{+}$
are studied in proton-proton collision data collected with the LHCb detector at
centre-of-mass energies of 7, 8, and 13 TeV, corresponding to a total
integrated luminosity of 9 fb$^{-1}$. The ratio of branching fractions of the
$B^{+}\to D^{*-}D^{+}_{s}\pi^{+}$ and $B^{0}\to D^{*-}D^{+}_{s}$ decays is
measured to be $0.173\pm 0.006\pm 0.010$, where the first uncertainty is
statistical and the second is systematic. Using partially reconstructed
$D^{*+}_{s}\to D^{+}_{s}\gamma$ and $D^{+}_{s}\pi^{0}$ decays, the ratio of
branching fractions between the $B^{+}\to D^{*-}D^{*+}_{s}\pi^{+}$ and
$B^{+}\to D^{*-}D^{+}_{s}\pi^{+}$ decays is determined as $1.31\pm 0.07\pm
0.14$. An amplitude analysis of the $B^{+}\to D^{*-}D^{+}_{s}\pi^{+}$ decay is
performed for the first time, revealing dominant contributions from known
excited charm resonances decaying to the $D^{*-}\pi^{+}$ final state. No
significant evidence of exotic contributions in the $D^{+}_{s}\pi^{+}$ or
$D^{*-}D^{+}_{s}$ channels is found. The fit fraction of the scalar state
$T_{c\bar{s} 0}^{\ast}(2900)^{++}$ observed in the $B^{+}\to
D^{-}D^{+}_{s}\pi^{+}$ decay is determined to be less than 2.3% at a 90%
confidence level.
