  Let $\mathcal T_\alpha~(0\leq\alpha<n)$ be a class of sublinear operators
satisfying certain size conditions introduced by Soria and Weiss, and let
$[b,\mathcal T_\alpha]~(0\leq\alpha<n)$ be the commutators generated by
$\mathrm{BMO}(\mathbb R^n)$ functions and $\mathcal T_\alpha$. This paper is
concerned with two-weight, weak type norm estimates for these sublinear
operators and their commutators on the weighted Morrey and amalgam spaces. Some
boundedness criterions for such operators are given, under the assumptions that
weak-type norm inequalities on weighted Lebesgue spaces are satisfied. As
applications of our main results, we can obtain the weak-type norm inequalities
for several integral operators as well as the corresponding commutators in the
framework of weighted Morrey and amalgam spaces.
