  Despite inherently poor interlayer conductivity, photodetectors made from
few-layer devices of 2D transition metal dichalcogenides (TMDs) such as WSe$_2$
and MoS$_2$ can still yield a desirably fast ($\leq$90 ps) and efficient
($\epsilon$$>$40\%) photoresponse. By combining ultrafast photocurrent (U-PC)
and transient absorption (TA) microscopy, the competing electronic escape and
recombination rates are unambiguously identified in otherwise complex kinetics.
Both the U-PC and TA response of WSe$_2$ yield matching interlayer electronic
escape times that accelerate from 1.6 ns to 86 ns with applied $E$-field to
predict the maximum device PC-efficiency realized of $\sim$44\%. The slope of
the escape rates versus $E$-field suggests out-of-plane electron and hole
mobilities of 0.129 and 0.031 cm$^2$/V$s$ respectively. Above $\sim$10$^{11}$
photons/cm$^{2}$ incident flux, defect-assisted Auger scattering greatly
decreases efficiency by trapping carriers at vacancy defects. Both TA and PC
spectra identify a metal-vacancy sub-gap peak with $\sim$5.6 ns lifetime as a
primary trap capturing carriers as they hop between layers. Synchronous TA and
U-PC microscopy show the\ net PC collected is modelled by a kinetic rate-law of
electronic escape competing against the linear and nonlinear Auger
recombination rates. This simple rate-model further predicts the PC-based
dynamics, nonlinear amplitude and efficiency, $\epsilon$ over a 10$^5$ range of
incident photon flux in few-layer WSe$_2$ and MoS$_2$ devices.
