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Boxy/peanut (BP) bulges are believed to originate from galactic discs through secular processes. A little explored question is how this evolution would be modified if the initial disc was assembled around a preexisting classical bulge. Previously we showed that a low-mass initial classical bulge (ICB), as might have been present in Milky Way-like galaxies, can spin up significantly by gaining angular momentum from a bar formed through disc instability. Here we investigate how the disc instability and the kinematics of the final BP bulge depend on the angular momentum of such a low-mass ICB. We show that a strong bar forms and transfers angular momentum to the ICB in all our models. However, rotation in the ICB limits the emission of the bar's angular momentum, which in turn changes the size and growth of the bar, and of the BP bulge formed from the disc.

The final BP bulge in these models is a superposition of the BP bulge formed via the buckling instability and the spun-up ICB. We find that the long-term kinematics of the composite BP bulges in our simulations is independent of the rotation of the ICB, and is always described by cylindrical rotation. However, as a result of the co-evolution between bulge and bar, deviations from cylindrical rotation are seen during the early phases of secular evolution and may correspond to similar deviations observed in some bulges. We provide a simple criterion to quantify deviations from pure cylindrical rotation, apply it to all our model bulges, and also illustrate its use for two galaxies: NGC 7332 and NGC 4570.