Conformance checking (CC) techniques of the process mining field gauge the conformance of the sequence of events in a case with respect to a business process model, which simply put is an amalgam of certain behavioral relations or rules. Online conformance checking (OCC) techniques are tailored for assessing such conformance on streaming events. The realistic assumption of having a finite memory for storing the streaming events has largely not been considered by the OCC techniques. We propose three incremental approaches to reduce the memory consumption in prefix-alignment-based OCC techniques along with ensuring that we incur a minimum loss of the conformance insights. Our first proposed approach bounds the number of maximum states that constitute a prefix-alignment to be retained by any case in memory. The second proposed approach bounds the number of cases that are allowed to retain more than a single state, referred to as multi-state cases. Building on top of the two proposed approaches, our third approach further bounds the number of maximum states that the multi-state cases can retain. All these approaches forget the states in excess to their defined limits and retain a meaningful summary of them. Computing prefix-alignments in the future is then resumed for such cases from the current position contained in the summary. We highlight the superiority of all proposed approaches compared to a state of the art prefix-alignment-based OCC technique through experiments using real-life event data under a streaming setting. Our approaches substantially reduce memory consumption by up to $80\%$ on average, while introducing a minor accuracy drop.