One could prove the following theorem in the smooth setting:
Theorem
Let $(M,m)$ be a $d$ dimensional $C^\infty$ manifold with smooth volume $m$. Let $\{F_i\}_{i=1}^k$ and $\{G_i\}_{i=1}^k$ be two systems of disjoint open subsets, satisfying $m(F_i)=m(G_i)$. Assume $\bar{F}_i$ and $\bar{G}_i$ are diffeomorphic to the $d$ dimensional closed ball in $\mathbb{R}^m$ for $i=1,\ldots k$. Then, given $\varepsilon>0$ and a compact set $N\subset M$ with $F_i\subset N$ for $i=1,\ldots , k$; there exists $h\in \text{Diff }^\infty(M,\mu)$ satisfying $m(h(F_i)\Delta G_i)<\varepsilon$.
This theorem is extremely helpful in building diffeomorphisms of smooth manifolds. I do not know a reference but I guess one can prove this using some kind of partition of unity argument.
My question is: What's the best one can do in the real-analytic set up? Of-course the entire business with $N$ needs to be dropped.
Also more generally: What are some general strategies of building real-analytic diffeomorphisms and flows?