In this paper, we derive long time Lp−Lq decay estimates, in the full range 1≤p≤q≤∞, for time-dependent multipliers in which an interplay between an oscillatory component and a diffusive component with different scaling appears. We estimate ‖m(t,⋅)‖Mpq as t→∞ for multipliers of typem(t,ξ)=e±i|ξ|σt−|ξ|θt, and suitable perturbations, under the assumption that the scaling of the diffusive component is worse, i.e., θ>σ. These multipliers are, for instance, related to the fundamental solution to the Cauchy problem for the σ-evolution equation with structural damping:utt+(−Δ)σu+(−Δ)θ2ut=0,t≥0,x∈Rn, in the so-called non-effective case σ<θ.