We are concerned with large-scale magnetic-field dynamo generation and propagation of magnetic fronts in turbulent electrically conducting fluids. An effective equation for the large-scale magnetic field is developed here that takes into account the finite correlation times of the turbulent flow. This equation involves the memory integrals corresponding to the dynamo source term describing the alpha effect and turbulent transport of magnetic field. We find that the memory effects can drastically change the dynamo growth rate, in particular, nonlocal turbulent transport might increase the growth rate several times compared to the conventional gradient transport expression. Moreover, the integral turbulent transport term leads to a large decrease of the speed of magnetic front propagation.