We introduce one-way flows in near algebras and two-way flows in double near algebras with two interrelated multiplications. We establish parametric representations of the one-way and two-way flows in terms of a single element of the algebra that we call a flow generator. We indicate probabilistic applications of the one-way flows to a study of polynomial stochastic processes. We apply our results on the two-way flows to harnesses and quadratic harnesses in probability theory, generalizing some previous results.