We introduce Onflow, a reinforcement learning method for optimizing portfolio allocation via gradient flows. Our approach dynamically adjusts portfolio allocations to maximize expected log returns while accounting for transaction costs. Using a softmax parameterization, Onflow updates allocations through an ordinary differential equation derived from gradient flow methods. This algorithm belongs to the large class of stochastic optimization procedures; we measure its efficiency by comparing our results to the mathematical theoretical values in a log-normal framework and to standard benchmarks from the 'old NYSE' dataset. For log-normal assets with zero transaction costs, Onflow replicates Markowitz optimal portfolio, achieving the best possible allocation. Numerical experiments from the 'old NYSE' dataset show that Onflow leads to dynamic asset allocation strategies whose performances are: a) comparable to benchmark strategies such as Cover's Universal Portfolio or Helmbold et al. ``multiplicative updates'' approach when transaction costs are zero, and b) better than previous procedures when transaction costs are high. Onflow can even remain efficient in regimes where other dynamical allocation techniques do not work anymore. Onflow is a promising portfolio management strategy that relies solely on observed prices, requiring no assumptions about asset return distributions. This makes it robust against model risk, offering a practical solution for real-world trading strategies.