Magnon transport through a magnetic insulator can be controlled by current-biased heavy-metal gates that modulate the magnon conductivity via the magnon density. Here, we report nonlinear modulation effects in 10nm thick yttrium iron garnet (YIG) films. The modulation efficiency is larger than 40%/mA. The spin-transport signal at high dc current density (2.2×1011A/m2) saturates for a 400nm wide Pt gate, which indicates that even at high current levels a magnetic instability cannot be reached in spite of the high magnetic quality of the films. </p