We discuss relatively complex and representative nonlinear fractional p-Laplace problems, in the setting of constant exponents and Kirchhoff terms. Two existence theorems of suitable solutions for a model problem with critical exponent are presented on the whole space \(\mathbb {R}^{n}\) . The Kirchhoff function of this problem may be zero at zero, and the lack of compactness in the critical case is the main feature and difficulty to overcome. Then, we deal with a critical fractional variable-order \(p(\cdot )\) -Kirchhoff type problem. We consider both the existence and asymptotic behavior of solution. The result is established combining appropriate continuous embeddings with the Palais–Smale property and a Brézis–Lieb type lemma. A dedicated framework setting is reviewed for a double nonlocal model problem, featured by two variable parameters. A particular attention is devoted to special continuous and compact embeddings about fractional Sobolev spaces with variable-order and variable exponent.