In this work, we present the first study addressing automatic speech recognition (ASR) for children in an online learning setting. This is particularly important for both child-centric applications and the privacy protection of minors, where training models with sequentially arriving data is critical. The conventional approach of model fine-tuning often suffers from catastrophic forgetting. To tackle this issue, we explore two established techniques: elastic weight consolidation (EWC) and synaptic intelligence (SI). Using a custom protocol on the MyST corpus, tailored to the online learning setting, we achieve relative word error rate (WER) reductions of 5.21 % with EWC and 4.36 % with SI, compared to the fine-tuning baseline.