Many real-world processes evolve in cascades over networks, whose topologies are often unobservable and change over time. However, the so-termed adoption times when for instance blogs mention popular news items are typically known, and are implicitly dependent on the underlying network. To infer the network topology, a dynamic structural equation model is adopted to capture the relationship between observed adoption times and the unknown edge weights, while accounting also for external (non-topological) perturbations. Assuming a slowly time-varying topology and leveraging the sparse connectivity inherent to social networks, edge weights are estimated by minimizing a sparsity-regularized exponentially-weighted least-squares criterion. To this end, a solver is developed by leveraging (pseudo) real-time sparsity-promoting proximal gradient iterations. Numerical tests with real cascades of online media demonstrate the effectiveness of the novel algorithm in unveiling sparse dynamically-evolving topologies.