This paper is concerned with the distributed finite-horizon filtering problem for a class of time-varying systems over lossy sensor networks. The time-varying system (target plant) is subject to randomly varying nonlinearities (RVNs) caused by environmental circumstances. The lossy sensor network suffers from quantization errors and successive packet dropouts that are described in a unified framework. Two mutually independent sets of Bernoulli distributed white sequences are introduced to govern the random occurrences of the RVNs and successive packet dropouts. Through available output measurements from not only the individual sensor but also its neighboring sensors according to the given topology, a sufficient condition is established for the desired distributed finite-horizon filter to ensure that the prescribed average filtering performance constraint is satisfied. The solution of the distributed filter gains is characterized by solving a set of recursive linear matrix inequalities. A simulation example is provided to show the effectiveness of the proposed filtering scheme.