  We define a Muckenhoup-type condition on weighted Morrey spaces using the
K\"othe dual of the space. We show that the condition is necessary and
sufficient for the boundedness of the maximal operator defined with balls
centered at the origin on weighted Morrey spaces. A modified condition
characterizes the weighted inequalities for the Calder\'on operator. We also
show that the Muckenhoup-type condition is necessary and sufficient for the
boundedness on weighted local Morrey spaces of the usual Hardy-Littlewood
maximal operator, simplifying the previous characterization of
Nakamura-Sawano-Tanaka. For the same operator, in the case of global Morrey
spaces the condition is necessary and for the sufficiency we add a local $A_p$
condition. We can extrapolate from Lebesgue $A_p$-weighted inequalities to
weighted global and local Morrey spaces in a very general setting, with
applications to many operators.
