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We study the kinematics of the giant H II regions NGC 5455 and NGC 5471 located in the galaxy M101, using integral field observations that include the Hbeta and [O III] 5007 emission lines, obtained with the Keck Cosmic Web Imager. We analyse the line profiles using both single and multiple Gaussian curves, gathering evidence for the presence of several expanding shells and moving filaments. The line decomposition shows that a broad (sigma = 30-50 km/s) underlying component is ubiquitous, extending across hundreds of pc, while a large fraction of the narrow components have subsonic line widths. The supersonic turbulence inferred from the global line profiles is consistent with the velocity dispersion of the individual narrow components, i.e. the global profiles likely arise from the combined contribution of discrete gas clouds. We confirm the presence of very extended (400 - 1200 km/s) low-intensity line components in three bright star-forming cores in NGC 5471, possibly representing kinematic signatures of supernova remnants. For one of these, the known supernova remnant host NGC 5471 B, we find a significantly reduced [O III]/Hbeta line ratio relative to the surrounding photoionized gas, due to the presence of a radiative shock at low metallicity. We explore the systematic width discrepancy between H I and [O III] lines, present in both global and individual spaxel spectra. We argue that the resolution of this long-standing problem lies in the physics of the line-emitting gas rather than in the smearing effects induced by the different thermal widths.