Answer Set Programming (ASP) is a declarative formalism, developed in the field of nonmonotonic reasoning and recognized as a powerful tool for Knowledge Representation and Reasoning. ASP features show potential also in the Stream Reasoning realm. Nevertheless, such scenario demands for repeated executions and requires reactive reasoning over rapidly changing data streams. Evaluating ASP programs from scratch at each time point represents a bottleneck. To overcome such limits, incremental reasoning techniques have been proposed. Overgrounding is an incremental grounding technique working under the answer set semantics that totally espouses the ASP declarative nature. Given a non-ground program to be repeatedly evaluated in consecutive time points over possibly differing sets of input facts, overgrounding maintains and enriches an overgrounded program, which lately converges to a propositional theory general enough to be reused together with possible future inputs. In this work, we focus on developments and extensions of overgrounding that could be beneficial in Stream Reasoning applications. In particular, we present some forgetting strategies purposely intended to mitigate the typical accumulation-oriented behavior of overgrounding by properly dropping accumulated atoms and/or rules.