Several primitive families in the inner region of the main asteroid belt were identified as potential sources for two near-Earth asteroids (NEAs), (101955) Bennu and (162173) Ryugu, targets of the sample-return missions OSIRIS-REx and Hayabusa2, respectively. Four of the families, located at high proper inclinations (i>10{deg}), have not yet been compositionally studied: Klio, Chaldaea, Chimaera, and Svea. We want to characterize and analyze these families within the context of our PRIMitive Asteroid Spectroscopic Survey (PRIMASS), in order to complete the puzzle of the origins of the two NEAs. We obtained visible spectra (0.5-0.9um) of a total of 73 asteroids within the Klio, Chaldaea, Chimaera, and Svea collisional families, using the instrument OSIRIS at the 10.4m Gran Telescopio Canarias. We performed a taxonomical classification of these objects, and an analysis of the possible presence of absorption bands related to aqueous alterations, comparing the results with already studied primitive families in the inner main belt. We present here reflectance spectra for 30 asteroids in the Klio family, 15 in Chaldaea, 20 in Chimaera, and 8 in Svea. We show that Klio, Chaldaea, and Chimaera members have moderately red spectral slopes, with aqueous alteration absorption bands centered around 0.7um, characteristic of the group of primitive families known as Erigone-like. In contrast, Svea shows no 0.7um features, and neutral and blue spectral slopes, and thus is a Polana-like family. While all four families might be related to (162173) Ryugu, the only family studied in this work that might be related to (101955) Bennu is Svea.