Non-$\mathrm{\forall }$-homogeneity in free groups

We prove that non-abelian free groups of finite rank at least 3 or of countable rank are not $\mathrm{\forall }$-homogeneous. We answer three open questions from Kharlampovich, Myasnikov, and Sklinos regarding whether free groups, finitely generated elementary free groups, and non-abelian limit groups form special kinds of Fra"iss'e classes in which embeddings must preserve $\mathrm{\forall }$-formulas. We also provide interesting examples of countable non-finitely generated elementary free groups.