Traditional continual learning methods prioritize knowledge retention and focus primarily on mitigating catastrophic forgetting, implicitly assuming that the data distribution of previously learned tasks remains static. This overlooks the dynamic nature of real-world data streams, where concept drift permanently alters previously seen data and demands both stability and rapid adaptation. We introduce a holistic framework for continual learning under concept drift that simulates realistic scenarios by evolving task distributions. As a baseline, we consider Full Relearning (FR), in which the model is retrained from scratch on newly labeled samples from the drifted distribution. While effective, this approach incurs substantial annotation and computational overhead. To address these limitations, we propose Adaptive Memory Realignment (AMR), a lightweight alternative that equips rehearsal-based learners with a drift-aware adaptation mechanism. AMR selectively removes outdated samples of drifted classes from the replay buffer and repopulates it with a small number of up-to-date instances, effectively realigning memory with the new distribution. This targeted resampling matches the performance of FR while reducing the need for labeled data and computation by orders of magnitude. To enable reproducible evaluation, we introduce four concept drift variants of standard vision benchmarks, where previously seen classes reappear with shifted representations. Comprehensive experiments on these datasets using several rehearsal-based baselines show that AMR consistently counters concept drift, maintaining high accuracy with minimal overhead. These results position AMR as a scalable solution that reconciles stability and plasticity in non-stationary continual learning environments. Full implementation of our framework and benchmark datasets is available at: github.com/AlifAshrafee/CL-Under-Concept-Drift.