  We present a comprehensive experimental study of magnetization and
magnetocaloric effect (MCE) in double perovskite (DP) materials $R_2$NiMnO$_6$
with $R =$ Pr, Nd, Sm, Gd, Tb, and Dy. While a paramagnetic to ferromagnetic
transition, with T$_{\rm C}$ in the range $\sim 100 - 200~$K, is a common
feature that can be attributed to the ordering of Mn$^{4+}$ and Ni$^{2+}$
magnetic moments, qualitatively distinct behavior depending on the choice of
$R$ is observed at low temperatures. These low-temperature anomalies in
magnetization are also manifest in the change in magnetic entropy, $-\Delta
S_{M}$, whose sign depends on the choice of $R$. In order to understand these
results, we present theoretical analysis based on mean-field approximation and
Monte Carlo simulations on a minimal spin model. The model correctly captures
the key features of the experimental observations.
