Authors’ abstract: In this paper, we prove some new characterizations of the weighted functions such that norm dynamic inequalities of mixed type involving operators of Hardy’s type with general kernels, of the form \[ \Vert\mathcal{H}_{\mathcal{K}}f\Vert_{\mathbb{L}_u^q([a,\infty)_{\mathbb{T}})}\leq A\Vert f\Vert_{\mathbb{L}_{\upsilon}^p([a,\infty)_{\mathbb{T}})}, \] hold for \(1<p\leq q<\infty\) and \(1<q<p<\infty\), where \(\mathcal{H}_{\mathcal{K}}f(x):=\int_a^{\sigma(x)}\mathcal{K}(\sigma(x),y) f(y)\,\varDelta y\) (here \(u\) and \(\upsilon\) are the weight functions). Corresponding results are also obtained for the adjoint operator \(\mathcal{H}_{\mathcal{K}}^{\ast}f(x):=\int_x^{\infty}\mathcal{K}(x,\sigma(y)) f(y)\,\varDelta y\), where \(\sigma(x)\) is the forward jump operator on time scales. Our results include some well known inequalities in the literature.