  In this paper we deal with the well-posedness of Dirichlet problems
associated to nonlocal Hamilton-Jacobi parabolic equations in a bounded, smooth
domain $\Omega$, in the case when the classical boundary condition may be lost.
We address the problem for both coercive and noncoercive Hamiltonians: for
coercive Hamiltonians, our results rely more on the regularity properties of
the solutions, while noncoercive case are related to optimal control problems
and the arguments are based on a careful study of the dynamics near the
boundary of the domain. Comparison principles for bounded sub and
supersolutions are obtained in the context of viscosity solutions with
generalized boundary conditions, and consequently we obtain the existence and
uniqueness of solutions in $C(\bar{\Omega} \times [0,+\infty))$ by the
application of Perron's method. Finally, we prove that the solution of these
problems converges to the solutions of the associated stationary problem as $t
\to +\infty$ under suitable assumptions on the data.
