We investigate theoretically field effect transistors based on single-walled carbon nanotubes (CNTFET) and explore two device geometries with suspended multiwalled carbon nanotubes (MWNT) functioning as gate electrodes. In the two geometries, a doubly or singly clamped MWNT is electrostatically deflected toward the transistor channel, allowing for a variable gate coupling and leading to, for instance, a superior subthreshold slope. We suggest that the proposed designs can be used as nanoelectromechanical switches and as detectors of mechanical motion on the nanoscale.