We have analyzed high spatial and spectral resolution Hα emission-line TAURUS data on two nearby giant H II regions, NGC 604 and NGC 588. Previous work shows that while NGC 604 is a relatively young system whose emission line kinematics is due to a combination of gravity and stellar wind-blown shells, NGC 588 seems more evolved, with its ionized gas kinematics dominated by large stellar wind shells. We show that for these two H II regions the distribution of points in the velocity dispersion (σ) vs intensity (I) diagram shows important differences probably due to their different evolutionary status. We suggest that this diagram is an excellent diagnostic to separate the main broadening mechanisms, i.e., those caused by shells and loops generated by massive stars, which finally lead to cloud dispersal, and gravitational broadening. We propose that the degree of evolution of giant H II regions, from formation to total dispersal of the ionized gas, can be tracked by means of their σ vs I diagrams.