Summary: In this paper, we show the existence of infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents. More precisely, we consider \[ \begin{cases} M([u]_{s(\cdotp)}^2)(-\Delta)^{s(\cdotp)}u + V(x)u = \lambda|u|^{p(x)-2}u + \mu |u|^{q(x)-2}u & \text{ in } \Omega, \\ u = 0 & \text{ in } \mathbb{R}^N \ \Omega, \end{cases} \] where \[ [u]_{s(\cdotp)} := \left( \int\int_{\mathbb{R}^{2N}} \frac{|u(x) -u(y)|^2}{|x-y|^{N+2s(x,y)}} dx \; dy \right)^{1/2}, \] \(N \geq 1, s(\cdotp): \mathbb{R}^N \times \mathbb{R}^N \to (0,1)\) is a continuous function, \( \Omega\) is a bounded domain in \(\mathbb{R}^N\) with \(N > 2s(x,y)\) for all \((x,y) \in \Omega \times \Omega, (-\Delta)^{s(\cdotp)}\) is the variable-order fractional Laplace operator, \(M: \mathbb{R}_0^+ \to \mathbb{R}_0^+\) and \(V: \Omega \to [0, \infty)\) are two continuous functions, \(\alpha, \beta > 0\) are two parameters and \(p,q \in C(\Omega)\) addition to using the new version of Clark’s theorem due to Liu and Wang to prove the existence of infinitely many solutions for the above problem, we also apply the symmetric mountain pass theorem, fountain theorem and dual fountain theorem to obtain the same conclusion. The main feature, as well as the main difficulty, of our problem is the fact that the Kirchhoff term \(M\) could be zero at zero.