  We apply the disc instability model for soft X-ray transients to identify
system parameters along evolutionary sequences of black hole X-ray binaries
(BHXBs) that are consistent with transient behaviour. In particular, we focus
on the hitherto neglected group of BHXBs with intermediate-mass giant donor
stars. These spend a significant fraction of their X-ray active phase crossing
the Hertzsprung gap.
  Three Case B binary sequences with a black hole accretor and 2.5 M_sun
initial donor mass are presented in detail. We formulate rules which summarize
the behaviour of these sequences and provide an approximate description for
Case B mass transfer in intermediate-mass BHXBs. Chiefly, the timescale of the
overall radius expansion is given by the initial donor mass, while the surface
appearance is determined by the current donor mass.
  With these rules we obtain a general overview of transient and persistent
behaviour of all intermediate-mass BHXBs by just considering single star
sequences of different mass. We find that although systems in the process of
crossing the Hertzsprung gap are in general persistently bright, with
Eddington- or super-Eddington transfer rates, there is a narrow instability
strip where transient behaviour is possible. This strip extends over a
secondary mass range 2.0-3.5 M_sun. GRO J1655-40 might be such a system. We
predict that there are no BHXB transients with donors more massive than 3.5
M_sun, and no neutron star transients in the Hertzsprung gap.
