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We present a new time-dependent multizone radiative transfer code and its first application to study the synchrotron self-Compton (SSC) emission of the blazar Mrk 421. The code couples Fokker–Planck and Monte Carlo methods in a two-dimensional (cylindrical) geometry. For the first time all the light traveltime effects (LTTE) are fully considered, along with a proper, full, self-consistent treatment of Compton cooling, which depends on them. We study a set of simple scenarios where the variability is produced by injection of relativistic electrons as a `shock front' crosses the emission region. We consider emission from two components, with the second component either being pre-existing and cospatial and participating in the evolution of the active region (background), or being spatially separated and independent, only diluting the observed variability (foreground). Temporal and spectral results of the simulation are compared to the multiwavelength observations of Mrk 421 in 2001 March. We find parameters that can adequately fit the observed SEDs and multiwavelength light curves and correlations. There remain, however, a few open issues, most notably (i) the simulated data show a systematic soft intraband X-ray lag, (ii) the quadratic correlation between the TeV γ-ray and X-ray flux during the decay of the flare has not been reproduced. These features turned out to be among those more affected by the spatial extent and geometry of the source, i.e. LTTE. The difficulty of producing hard X-ray lags is exacerbated by a bias towards soft lags caused by the combination of energy-dependent radiative cooling time-scales and LTTE. About the second emission component, our results strongly favour the scenario where it is cospatial and it participates in the flare evolution, suggesting that different phases of activity may occur in the same region. The cases presented in this paper represent only an initial study, and despite their limited scope they make a strong case for the need of true time-dependent and multizone modelling.