This paper addresses the asymptotics of functionals with linear growth depending on the Riesz s-fractional gradient on piecewise constant functions. We consider a general class of varying energy densities and, as s approaches one, we characterize their local limiting functionals in the sense of Gamma-convergence. This is based on joint works with Stefano Almi, Manuel Friedrich, and Francesco Solombrino. This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.