Constrained mixture models aim to describe growth and remodeling (G&R) processes in living tissues by accounting for the continuous turnover and adaptation of microstructurally defined constituents in response to individual growth stimuli. Despite their broad use, recent studies have reported numerical instabilities in finite element simulations based on such models. The origin and nature of these instabilities, however, are not yet fully understood. In this contribution, we systematically investigate numerical stability issues arising in different growth formulations that have been proposed in earlier works. In addition, possible strategies to improve numerical robustness are discussed, and the implications of the observed stability behavior are illustrated using an example of cardiac growth and remodeling under hypertensive loading. To this end, a von Neumann stability analysis is performed for elastic growth formulations, complemented by parameter studies with respect to the growth gain and the bulk modulus for inelastic growth formulations. For elastic growth, constraint-enforcing approaches based on augmented Lagrangian formulations are examined. While these methods improve the fulfillment of the growth constraint, they are found to worsen the stability behavior of the numerical scheme. In contrast, non-local growth formulations are considered, in which the growth stimulus is augmented by a diffusive exchange mechanism. Overall, the results emphasize that constrained mixture models exhibit conditional stability across different growth formulations. For elastic growth, stricter constraint enforcement via augmented Lagrangian methods reduces the stability margin, whereas non-local growth formulations increase numerical stability for both elastic and inelastic growth. These findings highlight that non-local growth laws can enhance numerical stability in homogenized constrained mixture models, which is essential for robust and predictive modeling across parameter regimes.