Let $\{(X, H, J_t)\}_{t\in (-\delta, \delta)}$ be a family of compact strongly pseudoconvex embeddable CR manifolds. Let $f_0: (X, H, J_0)\rightarrow \mathbb C^N$ be any CR embedding. A natural question asked by Burns-Epstein and Lempert is if we can find CR embeddings $f_t: (X, H, J_t)\rightarrow\mathbb C^N$ such that $f_t(X)$ are close to $f_0(X)$ as $t\approx0$. When ${\rm dim}X\geq 5$, Tanaka has already given an affirmative answer. However, Lempert and Catlin gave a counterexample when ${\rm dim}X=3$. In this talk, we will present two results related to this question. This talk is based on joint works with George Marinescu, Chin-Yu Hsiao and Guicong Su.