Natural complete problems in higher levels of the $\mathsf{W}$-hierarchy The $\mathsf{W}$-hierarchy is a hierarchy of complexity classes $\mathsf{W}[t]$ in parameterized complexity, see the Complexity Zoo for definitions. An alternative definition defines $\mathsf{W}[t]$ using weighted Fagin definability for $\Pi_t$-formulas of first-order logic, see the textbook by Flum and Grohe.

For the lowest classes $\mathsf{W}[1]$ and $\mathsf{W}[2]$, many natural complete problems are known, e.g. Clique and Independent Set are complete for $\mathsf{W}[1]$ , and Dominating Set and Hitting Set are complete for $\mathsf{W}[2]$, where each of these problems is defined as the corresponding well-known $\mathsf{NP}$-complete problem with the size of the required solution set as the parameter.   

Are there any known natural complete problems for classes higher up in the $\mathsf{W}$-hierarchy, in particular for $\mathsf{W}[3]$ and $\mathsf{W}[4]$?
I believe the title of this paper is self-explanatory and answers your question:
On product covering in 3-tier supply chain models: Natural complete problems for W[3] and W[4]