  We define positive Toeplitz operators between harmonic Bergman-Besov spaces
$b^p_\alpha$ on the unit ball of $\mathbb{R}^n$ for the full ranges of
parameters $0<p<\infty$, $\alpha\in\mathbb{R}$. We give characterizations of
bounded and compact Toeplitz operators taking one harmonic Bergman-Besov space
into another in terms of Carleson and vanishing Carleson measures. We also give
characterizations for a positive Toeplitz operator on $b^{2}_{\alpha}$ to be a
Schatten class operator $S_{p}$ in terms of averaging functions and Berezin
transforms for $1\leq p<\infty$, $\alpha\in\mathbb{R}$. Our results extend
those known for harmonic weighted Bergman spaces.
