The primordial non-Gaussianity parameters fNL and tauNL may be scale-dependent. We investigate the capability of future measurements of the CMB mu-distortion, which is very sensitive to small scales, and of the large-scale halo bias to test the running of local non-Gaussianity. We show that, for an experiment such as PIXIE, a measurement of the mu-temperature correlation can pin down the spectral indices n_fNL and n_tauNL to values of the order of 0.3 if fNL = 20 and tauNL = 5000. A similar value can be achieved with an all-sky survey extending to redshift z ~ 1. In the particular case in which the two spectral indices are equal, as predicted in models where the cosmological perturbations are generated by a single-field other than the inflaton, then the 1-sigma error on the scale-dependence of the non-linearity parameters goes down to 0.2.