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Simultaneous sublimation activity of four main-belt primitive asteroids (145 Adeona, 704 Interamnia, 779 Nina and 1474 Beira having considerable eccentricity of the orbits and passing near perihelion was discovered in 2012. It points likely to a widespread character of the phenomenon among primitive-type asteroids in the Main asteroid belt (MAB). Very probably, this is a sign of the presence of a considerable fraction of ices (predominately water ice) in the matter of such bodies. To investigate dynamical reasons of ice matter delivery to the MAB from behind of the “snow line” in the early Solar system, we modeled the migration of planetesimals and pre-planetary bodies from the feeding zones of Jupiter, Saturn, Uranus and Neptune to the MAB and calculated the probability of a collision of a planetesimal with icy content (probably about 1/3) with a parent asteroid body (PAB) moving in the MAB.