Deformation of finite regular languages Let $L \subseteq \{0,1\}^n$ be any finite regular language s.t it has an acyclic DFA. 
Let $C$ be some class of acyclic DFAs.

Let $\sigma \in S_n$ be a permutation on $n$ symbols. We can apply $\sigma$ to a string $w$ of length $n$ by permuting symbols of $w$ according to $\sigma$ and  we denote the new string by $\sigma(w)$.

Let $L(\sigma)=\{\sigma(w) \mid w \in L\}$.

What interesting classes $C$ one can define s.t for any finite regular language $L$, either $L$ has a DFA in the class $C$ (L is recognized by a DFA which satisfies properties of class C) or there is a permutation $\sigma$ such that $L(\sigma)$ has a DFA in the class $C$. 

Effect of permutation, an example: $L =\{ww | w \in \{0,1\}^n\}$ has only acyclic DFA of size $\Omega(2^n)$ but there is a $\sigma$ such that $L(\sigma) =\{w_1^2w_2^2\cdots w_n^2 | w \in \{0,1\}^n\}$ has $O(n)$ size DFA. I refer $L(\sigma)$ as a deformation of $L$.

Is there some references for study of these kind? are similar questions for CFL studied?

Thank you.
1) Here is a paper that studies your question in the case of finite languages $L$, and shows that if a finite language $L$ can be decided by an $n$-state DFA, then any permutation of $L$ can be decided by an $m$-state DFA for some $m \leq \frac13 (n^2 + n + 1)$:


  Palioudakis A., Cho DJ., Goč D., Han YS., Ko SK., Salomaa K. (2015) The State Complexity of Permutations on Finite Languages over Binary Alphabets. In: Shallit J., Okhotin A. (eds) Descriptional Complexity of Formal Systems. DCFS 2015. Springer Lecture Notes in Computer Science, vol 9118. 


See slides here.

2) You might be interested in studies of read-once (algebraic) branching programs. Technically, this is a setting where the order of the input matters, but many of the results in this area apply regardless of how the input is ordered. While ro(A)BPs are slightly stronger than DFAs, they are closely related.