Local scalar flame properties of freely propagating turbulent premixed flames including the flame curvature h, and local displacement speed relative to the fresh gases S-d(u), have been measured simultaneously for methane, propane, and hydrogen/air flames at Lewis numbers varying from 0.33 to 1.4 and ratios of rms turbulent velocity to unstretched laminar burning velocity u'/S-L, 0(u), varying from 0 to 3.1. Three different mixtures were separately spark-ignited in a vertical wind tunnel. The expanding flame freely propagated in a grid-generated decaying turbulent flow. An advanced field-imaging technique based on high-speed laser tomography measured the temporal evolution of local flame properties. Local flame curvature and local displacement speed were calculated from flame-front contours. Curvature probability density functions (pdfs) were negatively skewed, especially for nonunity Lewis numbers, and displacement speed distributions underlined the influence of local stretch and thermodiffusive effects on flame speed variations. The temporal evolution of the mean flamelet radius of curvature converges towards half of the integral length scale measured in the cold how. Flame response in terms of displacement speed to curvature is found to be statistically dependent, and a linear relationship is observed. For propane/air flames, the displacement speed can be assumed to be independent of local flame curvature at each stage of flame propagation, whereas a very strong increase of displacement speed with positive curvatures can be observed along the wrinkled flame contour for hydrogen/air flames.