This article is from the Model Rockets FAQ, by Wolfram von Kiparski with numerous contributions by others.
Basically, NO. Black powder booster motors will not ignite composite
motors. Therefore, you cannot use a composite upper stage in a traditional
multi-stage, black powder rocket. Also, there are no composite booster
motors currently in production. They all have delays (4 seconds being
the shortest current delay from Aerotech, for example) or are plugged.
Typically, you cannot (and should not) use these as boosters in standard
black-powder multi-staged rockets.
If you want to use composite motors in multi-stage models then you have to
use other methods of igniting the upper stage (whether black powder or
composite) than are used with black-powder-only rockets. One method is
to electronically ignite the upper stage motor using a mercury switch to
complete an electrical connection to a capacitor at first stage burn-out.
This, in turn, sets off a flash bulb/thermalite fuse combo which ignites
the upper stage motor. Another method is to ignite lengths of thermalite
fuse at the time the booster is ignited. The length of fuse determines
the delay before the upper stage is ignited. Refer to the 'Other Sources
of Information' section in Part 1 of the FAQ. The NCR High Power technical
reports on staging composite motors is applicable to multi-staged,
composite motor powered model rockets as well.
Bob Weisbe uploaded plans for a mercury switch-based staging system that he
used in a converted Estes Terrier-Sandhawk kit. The URL for these plans