We consider Dynamic Page Migration (DPM) problem, one of the  
fundamental subproblems of data management in dynamically changing networks. We investigate a hybrid scenario,
where access patterns to the shared object are dictated by an adversary, and each processor performs a random
walk in X. We extend the previous results of \cite{dynamic-page-migration}: we develop 
algorithms for the case where X is a ring, and prove that with high probability they achieve 
a competitive ratio of $\tilde{\O} (\min \{ \sqrt[4]{D}, n \})$, where $D$ is the size of the shared object
and $n$ is the number of nodes in the network. These results hold also for any 
d-dimensional torus or mesh with diameter at least $\tilde{\Omega}(\sqrt{D})$.