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In this study, we consider three dark energy models in which $\Lambda$ is notconstant, but has a dynamic nature that depends on the Hubble parameter $H$and/or its time derivative $\dot{H}$. We analyze the generalized running vacuummodel, for which $\Lambda(H)=A+BH^2+C\dot{H}$, along with the two modelsobtained by setting $B$ or $C$ equal to zero. A null value for $C$ yields theclassical running vacuum model (RVM), while $B=0$ corresponds to what we termthe generalized running vacuum sub-case, or GRVS. Our main aim is toinvestigate whether these models can accommodate non-zero spatial curvature. Tothis end, we carry out a Markov Chain Monte Carlo analysis using data for theobservables associated with Type-Ia supernovae, cosmic chronometers, the cosmicmicrowave background and baryon acoustic oscillations, as well as two valuesfor the Hubble constant. Then we include data relating to the growth oflarge-scale structure (LSS) and repeat the procedure. Our results indicate thattaking LSS observations into account helps to tighten constraints and determinea definite sign for the model parameters. In the case of the RVM and GRVS, theaddition of growth data results in dynamical vacuum energy being preferred to acosmological constant at a little over $1\sigma$. This happens in both the flatand non-flat scenarios -- there are only a few exceptions -- but comes at thecost of an extra parameter which can degrade the performance of the models (asassessed by model selection criteria). Of special relevance is the fact thatthe inclusion of LSS data appears to increase compatibility with a flatgeometry. It also brings the constraints on the Hubble constant closer to therange of values established by \emph{Planck}.

Endowing $\mathbf{\Lambda}$ with a dynamic nature: constraints in a spatially curved Universe