  We prove the convergence of the extremal processes for variable speed
branching Brownian motions where the "speed functions", that describe the
time-inhomogeneous variance, lie strictly below their concave hull and satisfy
a certain weak regularity condition. These limiting objects are universal in
the sense that they only depend on the slope of the speed function at $0$ and
the final time $t$. The proof is based on previous results for two-speed BBM
obtained in a recent paper of ours and uses Gaussian comparison arguments to
extend these to the general case.
