We study families $G(\lambda,\cdot)$ of entire functions that are approximated by a sequence of families $G_n(\lambda,\cdot)$ of entire functions, where $\lambda\in\C$ is a parameter. In order to control the dynamics, the families are assumed to be of the same constant finite type. In this setting we prove the convergence of the hyperbolic components in parameter space as kernels in the sense of Carathéodory.