  In the framework of the functional renormalization group and of the
perturbative, algebraic approach to quantum field theory (pAQFT), in [DDPR23]
it has been derived a Lorentian version of a flow equation \`a la Wetterich,
which can be used to study non linear, quantum scalar field theories on a
globally hyperbolic spacetime. In this work we show that the realm of validity
of this result can be extended to study interacting scalar field theories on
globally hyperbolic manifolds with a timelike boundary. By considering the
specific examples of half Minkowski spacetime and of the Poincar\'e patch of
Anti-de Sitter, we show that the form of the Lorentzian Wetterich equation is
strongly dependent on the boundary conditions assigned to the underlying field
theory. In addition, using a numerical approach, we are able to provide strong
evidences that there is a qualitative and not only a quantitative difference in
the associated flow and we highlight this feature by considering Dirichlet and
Neumann boundary conditions on half Minkowski spacetime.
