In this article we consider a natural class of random walks on free products of graphs, which arise as convex combinations of random walks on the single factors. From the works of Gilch [6,7] it is well-known that for these random walks the asymptotic entropy as well as the drift w.r.t. the natural transition graph distance and also w.r.t. the word length exist. The aim of this article is to formulate three central limit theorems with respect to both drift definitions and the entropy. In the case that the random walk depends on finitely many parameters we show that the corresponding variances in the central limit theorems w.r.t. both drifts vary real-analytically in terms of these parameters, while the variance in the central limit theorem w.r.t. the entropy varies real-analytically at least in the case of free products of finite graphs.