If Sunlike stars become a red giant and eventually a white dwarf, what do red dwarfs become? The Sun is said to become a red giant at the end of its life (before that it will become an orange subgiant first and then an orange giant or so) and after ejecting its outer layers it should become a white dwarf. If yellow dwarfs like the Sun become red giants, what do red dwarfs become? Even redder giants? How do the red giants of former red dwarfs (say Proxima Centauri) differ from those of former yellow dwarfs (say the Sun) and how would Proxima's white dwarf be different from the Sun's white dwarf? I mean, a red dwarf's eventual white dwarf would be much less massive than that of the Sun, right?
A relevant paper here is Laughlin, Bodenheimer &amp; Adams (1997) &quot;The End of the Main Sequence&quot;. From the abstract:

We find that for masses $M_\ast &lt; 0.25\  M_\odot$ stars remain fully
convective for a significant fraction of the duration of their evolution. The maintenance of full convection
precludes the development of large composition gradients and allows the entire star to build up a large
helium mass fraction. We find that stars with masses $M &lt; 0.20\ M_\odot$ will never evolve through a red giant
stage. After becoming gradually brighter and bluer for trillions of years, these late M dwarfs of today
will develop radiative-conductive cores and mild nuclear shell sources; these stars then end their lives as
helium white dwarfs.

Section 3 of the paper provides a detailed description of the lifetime of a $0.1\ M_\odot$ star. A brief summary:

After approximately 2 Gyr of contraction, the star reaches the zero-age main sequence point with a temperature of 2228 K and a luminosity of $10^{-3.38}\ L_\odot$.

On the main sequence, the mass fraction of $^3 \rm He$ increases steadily over a trillion years. The completely convective nature of the star ensures that it is mixed throughout the structure of the star. The star slowly increases its temperature and luminosity.

The maximum mass fraction of 9.95% $^3 \rm He$ is reached at 1380 Gyr. After this, the mass fraction declines as the rate of consumption exceeds the rate of production.

Between 1500 and 4000 Gyr (the text appears to use values that are too small by a factor of 1000 judging by figure 1 and the statement of total lifetime at the start of §3.2) the star starts turning itself into $^4 \rm He$, with this isotope becoming the main component of the star (by mass) around 3050 Gyr.

By 5740 Gyr, the star develops a radiative core due to the helium mass fraction lowering the opacity. This causes a small amount of contraction of the star and a decrease in luminosity.

After the development of the radiative core, shell burning proceeds outward through the star, increasing the surface temperature to a maximum of 5807 K at 6144 Gyr. The luminosity at this point is about $10^{-2.3}\ L_\odot$. This star is called a &quot;blue dwarf&quot;.

The star becomes cooler and less luminous. Shell burning continues during this time, eventually ending with the star having a hydrogen mass fraction of ~1%. The nuclear burning lifetime ends at 6281 Gyr, producing a helium white dwarf with temperature 1651 K and a luminosity of $10^{-5.287}\ L_\odot$.


A discussion of the appearance of the blue dwarf stage and how blue they actually are can be found in this question.
The $0.16 \le M_\ast / M_\odot \le 0.20$ range is transitional between the stars that become blue dwarfs and the stars that become red giants. From the paper:

In connection with their increased luminosity output, the
transitional stars in the mass range $0.16 \le M_\ast / M_\odot \le 0.20$
are able to produce increasingly larger expansions of the
overall stellar radius after the radiative hydrogen-exhausted
core has developed.

In the models calculated in the paper, the lowest mass object that unambiguously produced a red giant was $0.25\ M_\odot$, but as noted the transition region is not sharp. Nevertheless, this does mean that the higher-mass M dwarfs will eventually go through a red giant stage.