Long-term campaigns for
several blazars show that the radio emission occurs with a significant delay with respect to the  $\gamma$-ray band, with timescales ranging from weeks to years. This
observational evidence has long been a matter of debate and is usually
interpreted as a signature of the $\gamma$-ray emission originating upstream in the jet, with the emitting region becoming radio transparent at larger scales. In the presented analysis we show, by means of self-consistent numerical modelling, that a relativistic blob, undergoing an adiabatic expansion, can explain these delays, reproducing lags compatible with the observed timescales. We use the JetSeT framework to reproduce the numerical modelling of the radiative and accelerative processes, reproducing the temporal evolution of a single blob,
from the initial flaring activity and the subsequent expansion, following the spectral evolution and the corresponding light curves, investigating the relations among the observed parameters, rise time, delay, and decay time, and we identify the link with physical parameters. We find that, when adiabatic expansion is active, lags due to the shift of the synchrotron frequency occur. The delay, rising, and decaying timescales depends on the velocity of the expansion and on the time required for the source to exhibit an SSA frequency below the observed radio frequency. We derive an inter-band response function, embedding the parameters mentioned above, and we investigate the effects of the competition between radiative and adiabatic cooling timescales on the response. We apply the response function to long-term radio and  $\gamma$-ray light curves of Mrk 421, finding satisfactory agreement on the long-term behaviour, and we use a Monte Carlo Markov Chain approach to
estimate some relevant physical parameters.