MCG-6-30-15, at a distance of 37 Mpc (z=0.008), is the archetypical Seyfert 1 galaxy showing very broad Fe K$alpha$ emission. We present results from a joint NuSTAR and XMM-Newton observational campaign that, for the first time, allows a sensitive, time-resolved spectral analysis from 0.35 keV up to 80 keV. The strong variability of the source is best explained in terms of intrinsic X-ray flux variations and in the context of the light bending model: the primary, variable emission is reprocessed by the accretion disk, which produces secondary, less variable, reflected emission. The broad Fe K$alpha$ profile is, as usual for this source, well explained by relativistic effects occurring in the innermost regions of the accretion disk around a rapidly rotating black hole. We also discuss the alternative model in which the broadening of the Fe K$alpha$ is due to the complex nature of the circumnuclear absorbing structure. Even if this model cannot be ruled out, it is disfavored on statistical grounds. We also detected an occultation event likely caused by BLR clouds crossing the line of sight.