Learning multiple tasks sequentially requires neural networks to balance retaining knowledge, yet being flexible enough to adapt to new tasks. Regularizing network parameters is a common approach, but it rarely incorporates prior knowledge about task relationships, and limits information flow to future tasks only. We propose a Bayesian framework that treats the network's parameters as the state space of a nonlinear Gaussian model, unlocking two key capabilities: (1) A principled way to encode domain knowledge about task relationships, allowing, e.g., control over which layers should adapt between tasks. (2) A novel application of Bayesian smoothing, allowing task-specific models to also incorporate knowledge from models learned later. This does not require direct access to their data, which is crucial, e.g., for privacy-critical applications. These capabilities rely on efficient filtering and smoothing operations, for which we propose diagonal plus low-rank approximations of the precision matrix in the Laplace approximation (LR-LGF). Empirical results demonstrate the efficiency of LR-LGF and the benefits of the unlocked capabilities.