  In 2022 the DART mission spacecraft impacted the asteroid Dimorphos, the
secondary body of the binary Didymos system, ejecting a large number of dust
particles, rocks and boulders. The ESA Hera mission will reach the system in
2026 for post--impact studies and possible detection of orbiting fragments. We
investigate the long term dynamics of the large boulders ejected by DART to
test if any of these objects survive in orbit until the arrival of the Hera
mission. To model the dynamics of the boulders we use a numerical model which
includes the gravity of non-spherical Didymos and Dimorphos, the solar gravity
and the radiation pressure. The SPICE kernels are used to define the correct
reference frame for the integration. The dynamics of the boulders is highly
chaotic and 1% of the initial boulders survive at least for 4 years on
quasi--stable orbits. These orbits are characterised by wide oscillations in
eccentricity in antiphase with those in inclination (including spin flips), a
mechanism similar to the Kozai one. This behaviour may protect these bodies
from close encounters with both asteroids. We also compute the distribution on
the surfaces of the asteroids of sesquinary impacts which may influence the
dust emission, after the initial DART impact, and the surface composition of
the asteroids. The probability of observing boulders by the mission Hera is
small but not negligible and an almost constant flux of escaping boulders is
expected in the coming years since their lifetime after the DART impact covers
a large time interval. Most of re--impacts on Dimorphos occur in the hemisphere
opposite to the impact site, preferentially close to the equatorial plane.
