A partially turbulent flow continuously incorporates irrotational fluid into the turbulent region, a phenomenon known as entrainment. Although entrainment locally acts at viscous scales, the thin interface separating the turbulent from the irrotational region is extremely convoluted, and twisted in such a way that renders the global entrainment flux scale-independent. Despite turbulent entrainment being widely recognised as a multi-scale process, the theoretical basis for quantifying the entrainment flux at multi-scales is lacking. In this paper we derive an equation that allows us to quantify the local entrainment velocity at multi-scales. This is done by defining the local entrainment velocity as the propagation speed of an iso-surface of filtered enstrophy relative to the coarse-grained velocity field, and using the filtered enstrophy budget to split the total velocity into its individual components, i.e. viscous, inviscid, baroclinic and sub-filter. The equation is used to investigate the entrainment at multi-scales in simulated turbulent mixing layers, where turbulence is sustained by either a mean shear or an unstable buoyancy gradient (Rayleigh-Taylor turbulence).