  Binary systems of a hot subdwarf B (sdB) star + a white dwarf (WD) with
orbital periods less than 2-3 hours can come into contact due to gravitational
waves and transfer mass from the sdB star to the WD before the sdB star ceases
nuclear burning and contracts to become a WD. Motivated by the growing class of
observed systems in this category, we study the phases of mass transfer in
these systems. We find that because the residual outer hydrogen envelope
accounts for a large fraction of an sdB star's radius, sdB stars can spend a
significant amount of time ($\sim$10s of Myr) transferring this small amount of
material at low rates ($\sim 10^{-10}$-$10^{-9}\ M_\odot\,\rm yr^{-1}$) before
transitioning to a phase where the bulk of their He transfers at much faster
rates ($\gtrsim 10^{-8}\ M_\odot\,\rm yr^{-1}$). These systems therefore spend
a surprising amount of time with Roche-filling sdB donors at orbital periods
longer than the range associated with He star models without an envelope. We
predict that the envelope transfer phase should be detectable by searching for
ellipsoidal modulation of Roche-filling objects with $P_{\rm orb}=30$-$100$ min
and $T_{\rm eff}=20{,}000$-$30{,}000$ K, and that many ($\geq$10) such systems
may be found in the Galactic plane after accounting for reddening. We also
argue that many of these systems may go through a phase of He transfer that
matches the signatures of AM CVn systems, and that some AM CVn systems
associated with young stellar populations likely descend from this channel.
