  We prove that non-abelian free groups of finite rank at least 3 or of
countable rank are not $\forall$-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
$\forall$-formulas. We also provide interesting examples of countable
non-finitely generated elementary free groups.
