The evolution of a Taylor–Green forced magnetohydrodynamic system showing dynamo activity is analyzed via direct numerical simulations. The statistical properties of the velocity and magnetic fields in Eulerian and Lagrangian coordinates are found to change between the kinematic, nonlinear and saturated regime. Fluid element (tracer) trajectories change from chaotic quasi-isotropic (kinematic phase) to mean magnetic field aligned (saturated phase). The probability density functions (PDFs) of the magnetic field change from strongly nonGaussian in the kinematic to quasi-Gaussian PDFs in the saturated regime so that their flatness give a precise handle on the definition of the limiting points of the three regimes. Also the statistics of the kinetic and magnetic fluctuations along fluid trajectories changes. All this goes along with a dramatic increase of the correlation time of the velocity and magnetic fields experienced by tracers, significantly exceeding one turbulent large-eddy turn-over time. A remarkable consequence is an intermittent scaling regime of the Lagrangian magnetic field structure functions at unusually long time scales.