  Generalized circle packings were introduced in \cite{Ba-Hu-Sun} as a
generalization of tangential circle packings in hyperbolic background geometry.
In this paper, we introduce the combinatorial Calabi flow, fractional
combinatorial Calabi flow and combinatorial $p$-th Calabi flow for generalized
hyperbolic circle packings. We establish several equivalent conditions
regarding the longtime behaviors of these flows. This provides effective
algorithms for finding the generalized circle packings with prescribed total
geodesic curvatures.
