Suppose $\rho_1:G \rightarrow Diff(X)$ and $\rho_2:G\rightarrow Diff(X)$ are two smooth actions of a (not necessarily compact) Lie group $G$ on a smooth manifold $X$.
Suppose $Y_s$ for $s\in (-\epsilon,\epsilon)$ is a smooth family of submanifolds in $X$. Suppose moreover that both $\rho_1$ and $\rho_2$ preserve each $Y_s$ i.e. $\rho_i(g)(Y_s)\subset Y_s$ for all $g\in G$.
Finally, suppose that $\rho_1$ and $\rho_2$ are orbit equivalent along $Y_0$, that is, there exists a smooth function $\Phi:G \times Y_0 \rightarrow G$ such that $\rho_1(g)(y)=\rho_2(\Phi(g,y))(y)$ for all $g\in G$ and $y \in Y_0$. Are there conditions that guarantee one can extend this orbit equivalence to nearby $Y_s$?