I'm not sure what you mean by "exit region"... Typically, when one uses exit or entrance region to describe a jet feature, it's in terms of a jet streak (e.g. the left-exit region of the jet streak). For a straight upper-level jet streak, there is divergence (and assumed uperward motion) in the left-exit and right-entrance regions, and convergence (and assumed subsidence) in the right-exit and left-entrace regions. If a vort max is approaching, you could also see vertical motion owing to DIFFERENTIAL positive vorticity advection [again, PVA, or CVA, alone does not cause divergence or convergence and implied vertical motion -- you need to look at how the CVA/PVA changes with height].
So, let's say that there's a curved jet streak that's rounding the base of a progressive trough. In the left-exit region of that jet streak, there is upper-level divergence (and implied upward motion) from transverse circulation (it's the inertial-advective wind, technically); in the right-exit region, there is upper-level convergence (and implied downward motion). Now, if you add to that large-scale upward motion from DPVA, you could very well see enhanced upward motion in the left-exit region (up+up) and weaker downward motion (or neutral vertical motion) in the right-exit region (where the upward motion from DPVA may act to negate the downard motion from transverse circulation effects in the right-exit region of the jet streak). This can become very complex ,however, since there could be upward motion in the right-exit region of the curved jet streak downstream of a trough axis from warm-air advection as well (which isn't uncommon since there is a enhancement of the low-level flow towards the trough axis owing to the transverse circulation under the exit region of the jet streak).
For what it's worth, there is also, as noted previously, upper-level divergence maximized at the inflection point downstream of a trough axis and upstream of a ridge axis caused by ageostrophic curvature divergence aloft. As an air parcel moves through the base of the trough, it must slow owing to the cyclonic curvature, which means that the winds are subgeostrophic (imagine the total wind being the sum of the geostrophic wind and the ageostrophic wind, which means that, in this case, the ageostrophic wind acts against the geostrophic wind to make a new wind weaker than the geo wind). As a parcel moves through the top of a ridge, it speeds up, which means that winds are supergeostrophic (or, the ageostrophic wind is in the same direction as the geostrophic wind). By definition, the geostrophic wind is non-divergent, so we need to look at the ageostrophic wind for divergence/convergence (and implied vertical motion). If you can imagine, there's westerly ageo wind in the base of the trough, and easterly ageo wind at the top of the ridge (assuming a normal trough-ridge wavepattern in the NH) -- therefore, there is divergence of the ageostrophic wind between the ridge and the trough. On the other side of the trough (or to the west of the trough and east of a ridge in the NH), there is ageo convergence. This helps explain, in part (since thermal and vorticity patterns also significantly influence this), why the region downstream of a trough is a preferred region for surface cyclone development, and why the region downstream of a ridge is a preferred location for surface anticyclone development.
All of the above is in context of the quasi-geostrophic approximation. Ageostrophic responses become more important when more processes are included (i.e. in the semigeostrophic approximation).
