Let \(\alpha\geq-\frac12\), \(1<p<2\) and \(L_{p,\alpha}\) be the space of all \(f\) defined on \(\mathbb R\) such that \[ \|f\|_{p,\alpha}:=\left(\frac1{2^{\alpha+1}\Gamma(\alpha+1)}\int_{\mathbb R}|f(x)|^2|x|^{2\alpha+1}dx\right)^{1/p}<\infty. \] Let \(e_\alpha(x)=j_\alpha(x)+ic_\alpha xj_{\alpha+1}(x)\), \(c_\alpha=\frac1{2\alpha+2}\), \(i=\sqrt{-1}\), \(j_\alpha(\cdot)\) denotes the normalized Bessel function of the first kind. The Dunkl transform, \(\mathcal F_\alpha(f)(\lambda)=\frac1{2^{\alpha+1}\Gamma(\alpha+1)}\int_{\mathbb R}f(x)e_\alpha(\lambda x)|x|^{2\alpha+1}dx,\) turns out to be the Fourier transform when \(\alpha=-\frac12\). The authors derive two asymptotic formulas for \(\int_{|\lambda|\geq r}|\mathcal F_\alpha(f)(\lambda)|^q|\lambda|^{2\alpha+1}d\lambda\), \(r\rightarrow\infty\), \(\frac1p+\frac1q=1\). The first is proved when \(f\) belongs to what the authors call \((\beta,\gamma)\)-Dunkl Lipschitz class, while the second is given for functions of what the authors call \((p,\psi,\gamma)\)-Dini Lipschitz Dunkl class. The results extend known results in the case of the Fourier transform. The classes of functions mentioned above are defined in terms of generalized translation operators.