  We discuss in this paper the behaviour of minimal models of conformal theory
perturbed by the operator $\Phi_{13}$ at the boundary. Using the RSOS
restriction of the sine-Gordon model, adapted to the boundary problem, a series
of boundary flows between different set of conformally invariant boundary
conditions are described. Generalizing the "staircase" phenomenon discovered by
Al. Zamolodchikov, we find that an analytic continuation of the boundary
sinh-Gordon model provides a flow interpolation not only between all minimal
models in the bulk, but also between their possible conformal boundary
conditions. In the particular case where the bulk sinh-Gordon coupling is
turned to zero, we obtain a boundary roaming trajectory in the $c=1$ theory
that interpolates between all the possible spin $S$ Kondo models.
