Recently, an on-the-fly approach for (model-free) computational mechanics (DDCM) has been proposed and, among other applications, it has been used to accelerate FE^2 computations on the infinitesimal strain nonlinear elasticity context [1, 2]. Such a result is achieved by a simulation-specific online enrichment of the material database via computational homogenisation computations, which is shown to drastically reduce the number of necessary microscale solver calls compared to the standard FE^2 approach. Compared to other data-driven techniques, such as neural networks-based surrogate models, it is known that DDCM excels for sufficiently rich databases while delivering inaccurate results for poorly populated datasets [3]. Therefore, an efficient implementation of any active framework for DDCM should rely on reliable error estimators to guide simulation-specific database enrichments, associated with modified DDCM solvers properly dealing with sparse data. The extension of the aforementioned method to the finite strain regime, although straightforward, faces some challenges linked to the DDCM implementation in the finite strain regime, which is computationally less efficient. In this regard, while we can expect decreased speed-ups compared to the infinitesimal strain scenario, the method can still be efficient since the FE^2 finite strain computations are intrinsically more expensive, i.e., there is potentially more room for relieving the computational costs. Hence, the main goal of this talk is to better assess the suitability and the computational gains of the active approach for DDCM applied to the finite strain setting. After reviewing the most important methodological ingredients of the aforementioned approach, a set of meaningful numerical examples and speed-up comparisons will be presented. Implementation aspects of DDCM in the finite strain setting will be also discussed in the framework of ddfenics, an open-source FEniCSx-based DDCM implementation [4].