The primordial non-Gaussianity parameters fNL and τNL may be scale dependent. We investigate the capability of future measurements of the CMB μ 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 μ-temperature correlation can pin down the spectral indices nfNL and nτNL to values of the order of 0.3 if fNL=20 and τNL=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, the 1σ error on the scale dependence of the nonlinearity parameters goes down to 0.2.