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We present a theoretical assessment of the expected temporal rates of changeof periods ($\dot{\Pi}$) for low-mass ($M_{\star}/M_{\sun} \lesssim 0.45$) andextremely low-mass (ELM, $M_{\star}/M_{\sun} \lesssim 0.18-0.20$) white-dwarfstars, based on fully evolutionary low-mass He-core white dwarf and pre-whitedwarf models. Our analysis is based on a large set of adiabatic periods ofradial and nonradial pulsation modes computed on a suite of low-mass He-corewhite dwarf and pre-white dwarf models with masses ranging from $0.1554$ to$0.4352 M_{\sun}$. We compute the secular rates of period change of radial($\ell= 0$) and nonradial ($\ell= 1, 2$) $g$ and $p$ modes for stellar modelsrepresentative of ELMV and pre-ELMV stars, as well as for stellar objects thatare evolving just before the occurrence of CNO flashes at the early coolingbranches. We found that the theoretically expected magnitude of $\dot{\Pi}$ of$g$ modes for pre-ELMVs are by far larger than for ELMVs. In turn, $\dot{\Pi}$of $g$ modes for models evolving before the occurrence of CNO flashes arelarger than the maximum values of the rates of period change predicted forpre-ELMV stars. Regarding $p$ and radial modes, we found that the largerabsolute values of $\dot{\Pi}$ correspond to pre-ELMV models. We conclude thatany eventual measurement of a rate of period change for a given pulsatinglow-mass pre-white dwarf or white dwarf star could shed light about itsevolutionary status. Also, in view of the systematic difficulties in thespectroscopic classification of stars of the ELM Survey, an eventualmeasurement of $\dot{\Pi}$ could help to confirm that a given pulsating star isan authentic low-mass white dwarf and not a star from another stellarpopulation.

Pulsating low-mass white dwarfs in the frame of new evolutionary sequences: IV. The secular rate of period change