Context : An increasing number of low-mass ( M _ { \star } / M _ { \sun } \lesssim 0.45 ) and extremely low-mass ( ELM , M _ { \star } / M _ { \sun } \lesssim 0.18 - 0.20 ) white-dwarf stars are currently discovered in the field of the Milky Way .
Some of these stars exhibit long-period g -mode pulsations , and are called ELMV variable stars .
Also , some low-mass pre-white dwarf stars show short-period p -mode ( and likely radial-mode ) photometric variations , and are designated as pre-ELMV variable stars .
The existence of these new classes of pulsating white dwarfs and pre-white dwarfs opens the prospect of exploring the binary formation channels of these low-mass white dwarfs through asteroseismology .

Aims : We present a theoretical assessment of the expected temporal rates of change of periods ( \dot { \Pi } ) for such stars , based on fully evolutionary low-mass He-core white dwarf and pre-white dwarf models .

Methods : Our analysis is based on a large set of adiabatic periods of radial and nonradial pulsation modes computed on a suite of low-mass He-core white dwarf and pre-white dwarf models with masses ranging from 0.1554 to 0.4352 M _ { \sun } , which were derived by computing the non-conservative evolution of a binary system consisting of an initially 1 M _ { \sun } ZAMS star and a 1.4 M _ { \sun } neutron star companion .

Results : We compute the secular rates of period change of radial ( \ell = 0 ) and nonradial ( \ell = 1 , 2 ) g and p modes for stellar models representative of ELMV and pre-ELMV stars , as well as for stellar objects that are evolving just before the occurrence of CNO flashes at the early cooling branches .
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 are larger than the maximum values of the rates of period change predicted for pre-ELMV stars .
Regarding p and radial modes , we found that the larger absolute values of \dot { \Pi } correspond to pre-ELMV models .

Conclusions : We conclude that any eventual measurement of a rate of period change for a given pulsating low-mass pre-white dwarf or white dwarf star could shed light about its evolutionary status .
Also , in view of the systematic difficulties in the spectroscopic classification of stars of the ELM Survey , an eventual measurement of \dot { \Pi } could help to confirm that a given pulsating star is an authentic low-mass white dwarf and not a star from another stellar population .