One illustrative method disclosed herein includes forming a plurality of spaced-apart trenches in a semiconducting substrate to thereby define a fin structure for the device, forming a local isolation region within each of the trenches, forming a sacrificial gate structure on the fin structure, wherein the sacrificial gate structure comprises at least a sacrificial gate electrode, and forming a layer of insulating material above the fin structure and within the trench above the local isolation region. In this example, the method further includes performing at least one etching process to remove the sacrificial gate structure to thereby define a gate cavity, after removing the sacrificial gate structure, performing at least one etching process to form a recess in the local isolation region, and forming a replacement gate structure that is positioned in the recess in the local isolation region and in the gate cavity.