Separation profiles of free products

We deduce from a theorem of Dvorak–Norin that the separation and treewidth profiles of graphs are asymptotically equivalent, resolving a question of Huang–Hume–Kelly–Lam. As an application, we calculate the separation profiles of Cayley graphs of tree-graded graphs in terms of their pieces. Examples of tree-graded graphs include Cayley graphs of free products of finitely generated groups.