We study the n -th-order nonlinear dynamic equations x [ n ] ( t ) + p ( t ) ϕ α n − 1 [ ( x [ n − 2 ] ( t ) ) Δ σ ] + q ( t ) ϕ γ ( x ( g ( t ) ) ) = 0 on an unbounded time scale 𝕋 , where n ≥ 2 and for i = 1 , … , n − 1 x [ i ] ( t ) : = r i ( t ) ϕ α i [ ( x [ i − 1 ] ( t ) ) Δ ] , with r n = α n = 1 and x [ 0 ] = x ; here the constants α i and the functions r i , i = 1 , … , n − 1 , are positive and p , q are nonnegative functions. Criteria are established for the oscillation of solutions for both even- and odd-order cases. The results improve several known results in the literature on second-order, third-order, and higher-order linear and nonlinear dynamic equations. In particular our results can be applied when g is not (delta) differentiable and the forward jump operator σ and g do not commute.