  We show that for any uniformly bounded in time $H^1\cap L^1$ solution of the
dispersive generalized Benjamin-Ono equation, the limit infimum, as time $t$
goes to infinity, converges to zero locally in an increasing-in-time region of
space of order $t/\log t$. This result is in accordance with the one
established by Mu\~noz and Ponce \cite{MP1} for solutions of the Benjamin-Ono
equation. Similar to solutions of the Benjamin-Ono equation, for a solution of
the dispersive generalized Benjamin-Ono equation, with a mild $L^1$-norm growth
in time, its limit infimum must converge to zero, as time goes to infinity,
locally in an increasing on time region of space of order depending on the rate
of growth of its $L^1$-norm. As a consequence, the existence of breathers or
any other solution for the dispersive generalized Benjamin-Ono equation moving
with a speed "slower" than a soliton is discarded. In our analysis the use of
commutators expansions is essential.
