During laser fusion cutting, burr forms when the molten metal does not sufficiently exit the interaction zone. When it forms on the lower edge of the cut flank, burr becomes a factor limiting quality. Previous research has shown that a temporally regular and spatially localized melt flow can prevent the formation of burr. However, the high dynamics of the sub-processes involved can cause intrinsic instabilities that disrupt the flow and reduce the efficiency of the melt ejection. This paper presents a study on the correlation between process parameters, melt flow properties, and burr formation. It includes an experimental observation of the melt-flow dynamics using high-speed videography. In addition, a numerical CFD model was set up to examine fundamental flow properties, some of which are not observable experimentally. The dependency of the burr formation on the liquid Weber and Reynolds numbers is analyzed and it is demonstrated how the magnitude and allocation of vapor pressure gradients in the kerf decisively affect melt ejection and burr formation. Additionally, a previously unknown melt ejection regime is identified in the thick section range, which occurs at feed rates close to the maximum cutting speed under specific high-power process conditions. This regime is characterized by a significantly increased process efficiency that could open up a new high-speed process window.