The aim of this work is to study the compositional diversity of asteroid families based on their near-infrared colors, using the data within the MOVIS catalog. As of 2017, this catalog presents data for 53 436 asteroids observed in at least two near-infrared filters (Y, J, H, or Ks). Among these asteroids, we find information for 6299 belonging to collisional families with both Y -J and J-Ks colors defined. The work presented here complements the data from SDSS and NEOWISE, and allows a detailed description of the overall composition of asteroid families. We derived a near-infrared parameter, the ML*, that allows us to distinguish between four generic compositions: two different primitive groups (P1 and P2), a rocky population, and basaltic asteroids. We conducted statistical tests comparing the families in the MOVIS catalog with the theoretical distributions derived from our ML* in order to classify them according to the above-mentioned groups. We also studied the background populations in order to check how similar they are to their associated families. Finally, we used this parameter in combination with NEOWISE and SDSS to check for possible bimodalities in the data. We found 43 families with ML*err < 0.071 and with at least 8 asteroids observed: 5 classified as P1, 10 classified as P2, 19 families associated with the rocky population, and 9 families that were not linked to any of the previous populations. In these cases, we compared our samples with different combinations of these theoretical distributions to find the one that best fits the family data. We also show, using the data from MOVIS and NEOWISE, that the Bapistina family presents a two-cluster distribution in the near-infrared albedo vs. ML* parameter space that might be related to a common differentiated parent body. Finally, we show that the backgrounds we defined seem to be linked to their associated families.