We propose dynamics equations which describe the behaviour of non-stationary processes that follow the maximum Renyi entropy principle. The equations are derived on the basis of the speed-gradient principle originated in the control theory. The maximum of the Renyi entropy principle is analysed for discrete and continuous cases, and both a discrete random variable and probability density function (PDF) are used. We consider mass conservation and energy conservation constraints and demonstrate the uniqueness of the limit distribution and asymptotic convergence of the PDF for both cases. The coincidence of the limit distribution of the proposed equations with the Renyi distribution is examined.