  We prove a canonical bundle formula for generically finite morphisms in the
setting of generalized pairs (with $\mathbb{R}$-coefficients). This complements
Filipazzi's canonical bundle formula for morphisms with connected fibres. It is
then applied to obtain a subadjunction formula for log canonical centers of
generalized pairs. As another application, we show that the image of an
anti-nef log canonical generalized pair has the structure of a numerically
trivial log canonical generalized pair. This readily implies a result of
Chen--Zhang. Along the way we prove that the Shokurov type convex sets for
anti-nef log canonical divisors are indeed rational polyhedral sets.
