We consider the linear Dirac operator with a (<svg style="vertical-align:-0.0pt;width:18.65px;" id="M1" height="10.6875" version="1.1" viewBox="0 0 18.65 10.6875" width="18.65" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.6875)"> <g transform="translate(72,-63.45)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">−</tspan> <tspan style="font-size: 12.50px; " x="8.5645552" y="0">1</tspan> </text> </g> </g> </svg>)-homogeneous locally periodic potential that varies with respect to a small parameter. Using the notation of G-convergence for positive self-adjoint operators in Hilbert spaces we prove G-compactness in the strong resolvent sense for families of projections of Dirac operators. We also prove convergence of the corresponding point spectrum in the spectral gap.