By applying perturbed angular-correlation spectroscopy we have investigated spin correlations in the ``heavy-fermion'' compounds $\mathrm{U}({\mathrm{In}}_{1\ensuremath{-}x}{\mathrm{Sn}}_{x}{)}_{3},$ with $x=1.0,$ $0.7,$ and $0.5,$ by measuring the induced magnetic hyperfine field ${B}_{\mathrm{ind}}$ at diamagnetic ${}^{111}\mathrm{Cd}$ probe nuclei as a function of temperature and applied magnetic field. In zero applied field, the absence of any detectable magnetic hyperfine field at Cd reveals the absence of static magnetic correlations down to 4.2 K. However, from the field dependence of ${B}_{\mathrm{ind}}$ we find evidence for the presence of field-induced, short-ranged, and dynamic spin correlation between U f electrons at all compositions. The strength and dynamics of these correlated spins strongly depend on x, temperature, and applied magnetic field. As an important feature, for compositions near $x=0.5,$ classified as a heavy-fermion material with electronic specific heat coefficient $\ensuremath{\gamma}=500{\mathrm{m}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{}\mathrm{K}}^{2},$ these U spin correlations seem to set in from a relatively high temperature $(g~37$ K), and become very large on lowering temperature and/or increasing magnetic field, reflected in the measured Knight shift value K of about $\ensuremath{-}32%$ at 4.2 K and ${B}_{\mathrm{app}}=7$ T. We believe that these short-range spin correlations and their relaxation dynamics are responsible for the low temperature increase in magnetic susceptibility and electronic specific heat, previously considered to be an indication of heavy-fermion behavior in this system.