This paper deals with a class of nonlocal variable s(.)-order fractional
p(.)-Kirchhoff type equations: \begin{eqnarray*} \left\{
\begin{array}{ll}
\mathcal{K}\left(\int_{\mathbb{R}^{2N}}\frac{1}{p(x,y)}\frac{|\varphi(x)-\varphi(y)|^{p(x,y)}}{|x-y|^{N+s(x,y){p(x,y)}}}
\,dx\,dy\right)(-\Delta)^{s(\cdot)}_{p(\cdot)}\varphi(x) =f(x,\varphi)
\quad \mbox{in }\Omega,
\\ \varphi=0 \quad \mbox{on }\mathbb{R}^N\backslash\Omega. \end{array}
\right. \end{eqnarray*} Under some suitable conditions on the functions p,s,K and f, the existence and multiplicity of nontrivial solutions
for the above problem are obtained. Our results cover the degenerate case in
the p(β ) fractional setting