This work comprehensively describes and extends the results on asymptotic properties of linear discrete time-varying fractional order systems with Caputo and Riemann-Liouville forward and backward difference operators. In our considerations we take into account various definitions from the literature of fractional difference operators and we compare the dynamic properties of the corresponding systems. These equations are studied by converting them to the corresponding Volterra convolution equations. The main results are: explicit formulas for solutions, results on asymptotic stability, rates of growth or decay of solutions and solution separation. The work also formulates a number of open questions that may be the subject of future research.