Contrary to two-dimensional turbulence, decaying turbulence in a shallow fluid layer is characterized by the presence of long-lived meandering currents. This paper investigates the structure of vertical motion present inside the free-surface shallow-fluid layer and its relation with these currents. Downward motion is concentrated in thin elongated strain-dominated regions around instantaneous hyperbolic points and upward flow is observed in rotation-dominated patches around elliptic points. We show that the emergence of meandering currents results from inherent three-dimensionality of the flow due to the presence of the no-slip bottom. Our findings seriously question the assumption of quasi-two-dimensionality of shallow-fluid layer flows.