Lorentz-covariance of special-relativistic formulae expresses a definite spacetime symmetry, which is sometimes claimed to rule out the possibility of time flow. Such an argument from spacetime symmetry is strengthened by the general theory of relativity only if the general covariance of physical formulae yields more spacetime symmetry. It is argued that no general-relativistic spacetime is more symmetric than the special-relativistic one. The opposite opinion results from confusing dynamicity of spacetime metric with its relativity. Moreover, the whole argument from symmetry rests on confusing the physical spacetime with the full spacetime structure of the world. On the other hand, the so-called hole argument indicates some excess of generality in generally-covariant Einstein equations. The possibility of time flow is proposed as a selection rule for eliminating non-physical solutions. It is argued that this excludes no realistic cosmological models.