The paper determines criteria of elementary equivalence for some classes of free groups with operators and free products with the length function. The case of a group with operators admitting rational coordinatization with a finite basis is completely analyzed. They are polycyclic, solvable groups of finite rank without torsion, and Chernikov groups. The concept of $\omega$-isomorphism of groups intermediate between elementary equivalence and isomorphism is important for the aspects of elementary equivalence of groups with operators and free product. it is proved that $\omega$-isomorphism of arbitrary groups of operators is followed by the elementary equivalence of the respective free operator groups (free products) with length function.

Let $\kappa$, $\lambda$ be infinite cardinals, $\psi \in L_{\kappa^{+},\omega}$. We say that the sentence $\psi$ has the $\lambda$-unsuperstability property if there are $\{\varphi_{n}(\bar{\mathbf{x}},\bar{\mathbf{y}}): n < \omega\}$ quantifier free first order formulas in $L$, a model $M$ of $\psi$, and there exist $\{\bar{\mathbf{a}}_{\eta}: \eta \in^{\omega \geq} \lambda\} \subseteq |M|$ such that for all $\eta \in^{\omega}\lambda$, and for every $\nu \in^{\omega >}\lambda$, $$\nu < \eta \Leftrightarrow M \vDash \varphi_{l(\nu)}[\bar{\mathbf{a}}_{\nu},\bar{\mathbf{a}}_{\eta}].$$