We present approximation algorithms for some variants of center-based clustering and related problems in the fully dynamic setting, where the pointset evolves through an arbitrary sequence of insertions and deletions. Specifically, we target the following problems: $k$-center (with and without outliers), matroid-center, and diversity maximization. All algorithms employ a coreset-based strategy and rely on the use of the cover tree data structure, which we crucially augment to maintain, at any time, some additional information enabling the efficient extraction of the solution for the specific problem. For all of the aforementioned problems our algorithms yield $(α+\varepsilon)$-approximations, where $α$ is the best known approximation attainable in polynomial time in the standard off-line setting (except for $k$-center with $z$ outliers where $α= 2$ but we get a $(3+\varepsilon)$-approximation) and $\varepsilon>0$ is a user-provided accuracy parameter. The analysis of the algorithms is performed in terms of the doubling dimension of the underlying metric. Remarkably, and unlike previous works, the data structure and the running times of the insertion and deletion procedures do not depend in any way on the accuracy parameter $\varepsilon$ and, for the two $k$-center variants, on the parameter $k$. For spaces of bounded doubling dimension, the running times are dramatically smaller than those that would be required to compute solutions on the entire pointset from scratch. To the best of our knowledge, ours are the first solutions for the matroid-center and diversity maximization problems in the fully dynamic setting.