We characterize boundedness and compactness of weighted composition operators acting between weighted Bergman spaces $A_{v,p}$ and weighted Banach spaces $H_w^{\infty}$ of holomorphic functions on the open unit ball of $C^N$, $N\geq1$. Moreover, we give a sufficient condition for such an operator acting between weighted Bergman spaces $A_{v,p}$ and $A_{w,p}$ on the unit ball to be bounded.