  Motivated by surprises in recent experimental findings, we study transport in
a model of a quantum Hall edge system with a gate-voltage controlled
constriction. A finite backscattered current at finite edge-bias is explained
from a Landauer-Buttiker analysis as arising from the splitting of edge current
caused by the difference in the filling fractions of the bulk ($\nu_{1}$) and
constriction ($\nu_{2}$) quantum Hall fluid regions. We develop a hydrodynamic
theory for bosonic edge modes inspired by this model. The constriction region
splits the incident long-wavelength chiral edge density-wave excitations among
the transmitting and reflecting edge states encircling it. The competition
between two interedge tunneling processes taking place inside the constriction,
related by a quasiparticle-quasihole (qp-qh) symmetry, is accounted for by
computing the boundary theories of the system. This competition is found to
determine the strong coupling configuration of the system. A separatrix of
qp-qh symmetric gapless critical states is found to lie between the relevant RG
flows to a metallic and an insulating configuration of the constriction system.
This constitutes an interesting generalisation of the Kane-Fisher quantum
impurity model. The features of the RG phase diagram are also confirmed by
computing various correlators and chiral linear conductances of the system. In
this way, our results find excellent agreement with many recent puzzling
experimental results for the cases of $\nu_{1}=1/3,~1$. We also discuss and
make predictions for the case of a constriction system with $\nu_{2}=5/2$.
