[ShIfTiNgBoT]

Do not mix them (probably). There are two kinds of white lithium greases, a spray-on form, and a paste form. The spray on kind will not function as intended, as it evaporates and leaves a film behind which is the lubricant. You may have better luck with the paste form of white lithium (comes in tubes, not cans), although my guess for that is only speculation.

Either way, it's almost always good form to remove previous lubrication when switching to a different kind.