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A star does not turn into a red dwarf. A red dwarf is simply a star that has a low mass to begin with.
Red dwarf, yellow dwarf, red dwarf
A white dwarf could not become a red dwarf. A white dwarf is a remnant of a dead star. A red dwarf is a star with a very low mass.
Only a red dwarf star is red. Our Sun is a yellow dwarf.
A red dwarf is red because it is cool, and cool colours are red whereas hot colours are white and blue.
Proxima centauri is the closest red dwarf star and is the closest star