Simple property of DFAs
Actually I think I have spotted this property in a book. It uses the "extended" transition relation $\delta^*$, i.e., for strings rather than single letters. 


  Other properties you would expect $\delta^*$ to satisfy can be derived from
  our definition. For example, a natural generalization of the recursive
  statement in the definition is the formula 
  $$\delta^*(q,xy) = \delta^*(\delta^*(q,x),y))$$ 
  which is true for every state $q$ and every two strings $x$ and $y$ in $\Sigma^*$.
  The proof is by
  structural induction on $y$ and is similar to the proof [...] in Example 1.27.
  
  John C. Martin, Introduction to Languages and the Theory of Computation, McGraw-Hill, 4th edition, c2011. Page 54.


I hope you agree with me that this matches the "machine" terminology you use in your question.