Long-time behavior for systems of Fisher-KPP type with interacting components

We study the long-time behavior of a triangular system of Fisher--KPP type with $k$ interacting components, associated with a reducible multitype branching Brownian motion with $k$ types of particles. For this cascading system, we prove convergence in shape of each component to the minimal-speed Fisher--KPP traveling wave and determine the front asymptotics up to the constant order. This yields a PDE proof of Conjecture 1.2 from [4] on the convergence in distribution of the centered maximum particle in a cascading branching Brownian motion. We also derive asymptotic front-location estimates for such systems with general Fisher--KPP nonlinearities.