  In this paper, the fractional Hardy-type operator of variable order
$\beta(x)$ is shown to be bounded from the variable exponent Herz-Morrey spaces
$M\dot{K}_{p_{_{1}},q_{_{1}}(\cdot)}^{\alpha(\cdot),\lambda}(\R^{n})$ into the
weighted space
$M\dot{K}_{p_{_{2}},q_{_{2}}(\cdot)}^{\alpha(\cdot),\lambda}(\R^{n},\omega)$,
where $\alpha(x)\in L^{\infty}(\mathbb{R}^{n})$ be log-H\"older continuous both
at the origin and at infinity, $\omega=(1+|x|)^{-\gamma(x)}$ with some
$\gamma(x)>0$ and $ 1/q_{_{1}}(x)-1/q_{_{2}}(x)=\beta(x)/n$ when $q_{_{1}}(x)$
is not necessarily constant at infinity.
