I think this question is too broad, but I'll take a stab at it.
The Russell-Vogt (or sometimes Vogt-Russell) theorem is that the position of a star in the HR diagram is determined by its mass and composition.
The luminosity is determined mostly by its central temperature and composition. In turn, the central temperature depends on mass and radius and the radius depends on the luninosity and effective temperature.
Thus the question you ask fills textbooks. But to first order.
The time dependence of luminosity is set by the time dependence of mass - ie mass loss (or gain) and the rate of change of composition, particularly in the nuclear burning regions of a star.
In a star like the Sun, mass loss is relatively unimportant, so it is the rate at which hydrogen is turned into helium in the core that sets the timescale for luminosity evolution. Other processes that alter the core composition like mixing (due to convection, rotational mixing or diffusion) are thought to be second order effects.
Restricting myself to the main sequence as an example. The luminosity of the Sun is set by the core temperature. The the nuclear reactions become faster as the mean particle weight increases and the core contracts and the temperature rises to maintain hydrostatic equilibrium. The rate at which this occurs is given by the luminosity of the star divided by the amount of H available (proportional to the mass of the star).
Overall chemical composition drives this as a second order effect. A lower metallicity star is less opaque to radiation. The radiation escapes more efficiently and the star is smaller and hotter at the surface for the same luminosity. Smaller stars are hotter in the middle for the same mass and thus more luminous. Thus the luminosity evolution of low metallicity stars is faster than those of higher metallicity. See for example Figs 1 and 2 of Bazan & Mathews (1990).
