By asking how far mountain lions travel I am presuming that the question means how far pumas travel while hunting. In general terms they travel extensively. A scientist, Seidensticker, reports that pumas ‘were constantly moving through the country in a way that optimised encounters with prey…’. Males travel further than females. Females can be limited by the age of their offspring.
There’s not much information on actual distances travelled during daily activities. One report by a scientist, Musgrave, states that an old male puma covered more than 32 kilometres in a single night (20 miles). The cat travelled along a high ridge and over peaks. It has been reported that two adult males in Mexico travelled 5.2 and 7.9 kilometres in a day. Four females with with small kittens travelled 1, 1.2, 2.5 and 3.1 kilometres respectively in one day.
In the Santa Ana Mountains in Southern Calfornia pumas travelled 6.4 kilometres per 24 hours. Most travelling occurred at night.
Source: Wild Cats of the World by the Sunquists.