<p>Projection-based reduced-order models (ROMs) are typically deployed as static surrogates, which limits their practical utility once the system leaves the training manifold. In this work, we formalize and study adaptive non-intrusive ROMs that update both the latent subspace and the reduced dynamics online. Building on ideas from static non-intrusive ROMs, specifically, operator inference (OpInf) and the recently introduced non-intrusive trajectory-based optimization of reduced-order models (NiTROM), we propose three adaptive formulations: adaptive OpInf, which performs sequential basis and operator refits; adaptive NiTROM, which jointly optimizes the encoder, decoder, and polynomial dynamics via Riemannian optimization; and a hybrid approach that initializes NiTROM with an OpInf update. We analyze the structure and computational cost of these adaptive models, including the role of the lookback data window, adaptation frequency, and optimization budget. The methods are evaluated on a transiently perturbed lid-driven cavity flow, where static ROMs drift or destabilize when forecasting beyond the training regime. In contrast, adaptive OpInf can robustly suppress amplitude drift at modest cost, while adaptive NiTROM can attain near-exact energy tracking under frequent updates but remains sensitive to initialization and optimization depth. The hybrid method proves most reliable under regime changes and limited offline data, producing physically coherent fields and bounded energy over the prediction horizons considered. We argue that predictive claims for ROMs must be cost-aware and transparent, with clear separation of training/adaptation/deployment regimes and explicit reporting of online budgets. Overall, this work provides a practical template for building adaptive non-intrusive ROMs that incorporate online corrections and can extend predictive capability beyond the initial training manifold.</p>