  We consider the two-point functions of conserved bulk currents and
energy-momentum tensor in a boundary CFT defined on $\mathbb{R}_-^{1,2}$.
Starting from the consistent forms of boundary gauge and gravitational
anomalies we derive their respective contributions to the correlation functions
in the form of anomalous Ward identities. Using the recently developed momentum
space formalism we find an anomalous solution to each of these identities
depending on a single undetermined form-factor. We study the solution in two
different kinematic limits corresponding to small and large momentum $p_n$,
perpendicular to the boundary. We find that the anomalous term interpolates
between a non-local form resembling the standard anomaly-induced term in a
two-dimensional CFT at small $p_n$ and Chern-Simons contact terms at large
$p_n$. Using this we derive some consistency conditions regarding the
dependence of these anomalies on the boundary conditions and discuss possible
cancellation mechanisms. These ideas are then demonstrated on the explicit
example of free, massless three-dimensional fermion. In particular we manage to
obtain the respective anomalies via a diagrammatic momentum space computation
and expose the well-known relation between bulk parity anomaly and boundary
gauge anomalies.
