In this paper, the fractional Hardy-type operator of variable order ${eta}(x)$ is shown to be bounded from the Herz-Morrey spaces $Mdot{K}^{{alpha},{lambda}}_{p_1,q_1({cdot})}(mathbb{R}^n)$ with variable exponent $q_1(x)$ into the weighted space $Mdot{K}^{{alpha},{lambda}}_{p_2,q_2({cdot})}(mathbb{R}^n,{omega})$, where ${omega}=(1+|x|)^{-{gamma}(x)}$ with some ${gamma}(x)$ > 0 and $1/q_1(x)-1/q_2(x)={eta}(x)/n$ when $q_1(x)$ is not necessarily constant at infinity. It is assumed that the exponent $q_1(x)$ satisfies the logarithmic continuity condition both locally and at infinity that 1 < $q_1({infty}){leq}q_1(x){leq}(q_1)+$ < ${infty}(x{in}mathbb{R}^n)$.