Time series of images of a sunspot, acquired simultaneously in blue light and in G-band at the SVST, La Palma, are used to study horizontal motions of granules in the vicinity of the spot and evolution of penumbral grains (PGs) near the outer penumbral border. Local correlation tracking and feature tracking algorithms have been utilized for this purpose. Around sunspots and pores we can observe numerous centers of diverging local horizontal motions, caused mostly by exploding granules. Around developed spots we find an organized motion directed out from the penumbra. The divergence centers are carried away from the spot by this motion. In the outer penumbra, PGs move mostly outwards, toward the surrounding granulation. About 2/3 of PGs disappear near the penumbra-granulation border. The remaining PGs move across the border, they transform either to granules or to small bright features, and continue moving away from the sunspot.