In this paper we present a tatonnement process describing adaptations of prices and quantities in a convex general equilibrium model with production. The adaptations are determined by the state of the market and the starting price vector. The main result of the paper shows that for any economy out of a broad class of so-called, semi-algebraic convex economies and for any starting tuple consisting of a price vector and a related demand and supply, the process defines at least one path connecting the starting tuple and an equilibrium. The second result concerns the uniqueness of the path, given an arbitrary starting tuple, for a generic economy out of most standard classes of convex production economies, with consumer demand being unique at all price vectors. In that way, we generalize existing results on converging processes.