We investigate driven magnetohydrodynamic (MHD) turbulence by including the effects of expansion and collapse of background medium. The main goal is to quantify the evolution and saturation of strength and characteristic lengths of magnetic fields in expanding and collapsing media. Our findings are as follows. First, with expansion and collapse of background medium, the magnetic energy density per comoving volume does not saturate; either it keeps decreasing or increasing with time. The magnetic energy density relative to the kinetic energy density strongly depends on the expanding or collapsing rate. Second, at scales close to the energy injection (or driving) scale, the slope of magnetic field power spectrum shallows with expansion but steepens with collapse. Third, various characteristic lengths, relative to the energy injection scale, decrease with expansion but increase with collapse. We discuss the astrophysical implications of our findings.