  Systems of integral equations are proposed which generalise those previously
encountered in connection with the so-called staircase models. Under the
assumption that these equations describe the finite-size effects of
relativistic field theories via the Thermodynamic Bethe Ansatz, analytical and
numerical evidence is given for the existence of a variety of new roaming
renormalisation group trajectories. For each positive integer $k$ and
$s=0,\dots, k-1$, there is a one-parameter family of trajectories, passing
close by the coset conformal field theories $G^{(k)}\times
G^{(nk+s)}/G^{((n+1)k+s)}$ before finally flowing to a massive theory for
$s=0$, or to another coset model for $s \neq 0$.
