About the equilibrium of a flowing reaction

Imagine to have a tube (to make a reactor simple) inside which a chemical reactions take place and reactants are constantly fed so overall can be considered in steady-state conditions. 


Just to pick one, I assume the reaction to be exothermic.


Now in my mind there are the following two options, one of them is not clear.


1. Assuming that the tube is cooled and kept to a constant temperature and that the reaction is not controlled by kinetics, I think that at the end of the tube (if long enough) it can be considered in thermodynamic equilibrium.
2. If the tube is adiabatic then a distribution of temperature inside should arise. For each temperature there exists an equilibrium constant, and thus a different composition. In this case, would there be thermodynamic equilibrium at each single step (infinitesimal) along the tube? Again, this is considering an equilibrium-controlled reaction.


