I suspect the following statement is true:
Let $(M^3,g)$ be a compact and orientable Riemannian $3$-manifold with nonempty boundary. Let $\{\Sigma_n\}_{n \geq 1}$ be a sequence of compact and orientable free boundary minimal surfaces embedded in $M$. Suppose that there are positive constants $C_0$ and $C_1$ such that
(i) $\sup_{x \in \Sigma_n} \Vert A_{\Sigma_n}(x) \Vert^2 \leq C_0$ for every $n \geq 1$, where $A_{\Sigma_n}(x)$ denotes the second fundamental form of $\Sigma_n$ at the point $x$;
(ii) $\operatorname{Area}(\Sigma_n) \leq C_1$ for every $n \geq 1$.
Then there exists a subsequence $\{ \Sigma_{n_k} \}_{k \geq 1}$ of $\{ \Sigma_n \}_{n \geq 1}$ that converges smoothly and locally uniformly to an embedded free boundary minimal surface $\Sigma_{\infty} \subset M$.
Do you have a possible reference for this result?