The discovery of the Fermi bubbles - a huge bilobular structure seen in GeV gamma-rays above and below the Galactic center - implies the presence of a large reservoir of high energy particles at $\sim 10 \, \text{kpc}$ from the disk. The absence of evidence for a strong shock coinciding with the edge of the bubbles, and constraints from multi-wavelength observations point towards stochastic acceleration by turbulence as a likely model of acceleration. We have investigated the time-dependent acceleration of electrons and protons in a large-scale outflow from the Galactic centre. For the first time, we present a detailed numerical solution of the transport equation that includes the relevant transport and energy loss processes. Fitting to the observed spectrum and morphology of the bubbles allows determining the parameters of the outflow, thereby shedding light on the origin of the Fermi bubbles.