  This paper deals with the periodic homogenization of nonlocal parabolic
Hamilton-Jacobi equations with superlinear growth in the gradient terms. We
show that the problem presents different features depending on the order of the
nonlocal operator, giving rise to three different limit problems. To prove the
locally uniform convergence to the unique solution of the Cauchy problem for
the effective equation we need a new comparison principle among viscosity
semi-solutions of integro-differential equations that can be of independent
interest.
