Most studies investigating models and algorithms for distributed constraint optimization problems (DCOPs) assume messages arrive instantaneously or within a (short) bounded delay. Specifically, distributed local search DCOP algorithms have been designed as synchronous algorithms, performing in an asynchronous environment, i.e., algorithms that perform in synchronous iterations in which each agent exchanges messages with all its neighbors. This is true also for an anytime mechanism that reports the best solution explored during the run of synchronous distributed local search algorithms. Thus, when the assumptions on instantaneous message arrival are relaxed, the state of the art local search algorithms and mechanism do not apply. In this work, we address this limitation by: (1) Investigating the performance of existing local search DCOP algorithms in the presence of message latency; (2) Proposing an asynchronous monotonic distributed local search DCOP algorithm; and (3) Proposing an asynchronous anytime framework for reporting the best solution explored by non-monotonic asynchronous local search DCOP algorithms. Our empirical results demonstrate that, up to some extent, message delays have a positive effect on distributed local search algorithms due to increased exploration. The asynchronous anytime framework proposed, allows a maximal benefit from such latency based explorative heuristics.