We establish the existence of a weak solution for a Dirichlet fractional Kirchhoff–type problem driven by a nonlocal Musielak operator with genuinely (x, y)–dependent growth on a bounded Lipschitz domain \(\Omega \subset \mathbb {R}^{N}\) . The Kirchhoff coefficient depends on the global fractional Musielak modular, while the source term is merely Carathéodory with subcritical growth and is not assumed to be of potential type. The proof is nonvariational: we construct Faedo–Galerkin approximations in the nonlocal Dirichlet space \(W^{s}_{0}L_{\Phi _{x,y}}(Q)\) and handle the \(\sigma \) –finite interaction set Q by a Young–measure representation on a finite–measure exhaustion combined with a diagonal extraction. Assuming uniform convexity of the Musielak density, we derive an \((S_+)\) –type criterion, obtain strong convergence in the energy space, and pass to the Kirchhoff factor while identifying the nonlocal nonlinear term. The result applies in particular to variable–exponent, double–phase, and logarithmically perturbed kernels.
