In laser cutting, the fundamental role of the gas flow for melt removal and kerf formation is generally accepted. Beyond this vague understanding, however, the underlying physical mechanisms are not yet fully understood. In particular, detailed data concerning the momentum and heat transfer between the gas and melt have seldom been reported. This study addresses the local interactions between the cutting gas and kerf surface (melt film surface) in a fundamental way based on a combined experimental, theoretical, and numerical approach. Typical solid-state laser cut edges are analyzed considering the characteristic surface structures and the basic influences of the gas flow on the global and local melt movement. Here, apparent structures in the micrometer range indicate the effect of vortical gas structures close to the wall. Theoretical investigation of the gas boundary layer is conducted by semi-empirical equations and the transfer of basic results from boundary layer theory. It is shown that the boundary layer is in transition between laminar and turbulent flow, and local flow separations and shock-boundary layer interactions primarily induce spatially periodic and quasi-stationary instability modes. An improved numerical model of the cutting gas flow confirms the theoretical results and exhibits good agreement with experimental cut edges, reproducing the relevant instability modes and quantifying the local momentum and heat transfer distributions between gas and melt. With the knowledge gained about the underlying physical mechanisms, promising approaches for improvements of the fusion cutting performance are proposed.