  A secular variation of $G$ modifies the structure and evolutionary time
scales of white dwarfs. Using an state-of-the-art stellar evolutionary code, an
up-to-date pulsational code, and a detailed population synthesis code we
demonstrate that the effects of a running $G$ are obvious both in the
properties of individual white dwarfs, and in those of the white dwarf
populations in clusters. Specifically, we show that the white dwarf
evolutionary sequences depend on both the value of $\dot G/G$, and on the value
of $G$ when the white dwarf was born. We show as well that the pulsational
properties of variable white dwarfs can be used to constrain $\dot G/G$.
Finally, we also show that the ensemble properties of of white dwarfs in
clusters can also be used to set upper bounds to $\dot G/G$. Precisely, the
tightest bound --- $\dot G/G \sim -1.8 10^{-12}$ yr$^{-1}$ --- is obtained
studying the population of the old, metal-rich, well populated, open cluster
NGC 6791. Less stringent upper limits can be obtained comparing the theoretical
results obtained taking into account the effects of a running $G$ with the
measured rates of change of the periods of two well studied pulsating white
dwarfs, G117--B15A and R548. Using these white dwarfs we obtain $\dot G/G\sim
-1.8\times 10^{-10}$ yr$^{-1}$, and $\dot G/G\sim -1.3\times 10^{-10}$
yr$^{-1}$, respectively, which although less restrictive than the previous
bound, can be improved measuring the rate of change of the period of massive
white dwarfs.
