In the context of vessels of honor and dishonor, there are Godly people and ungodly people. We will speak more about ungodly people a little later. Paul is saying in Galatians 5 that with Godly people, we are in a constant battle to walk in the Spirit.
Peter, he had a very public moment of gratifying the flesh which is found in all four of the gospels - he denied Jesus.
Paul persecuted the disciples until He met Jesus on the road to Damascus.
King Saul had his heart changed by God but he turned to his own way walking in the flesh. God rejected him as King and he was killed by the Philistines along with his sons on Mount Gilboa.
King David walked in the flesh by commiting adultery with Bathsheba and then having her husband, Uriah, killed. Before that, he tried to become an ally of the Philistines.
The Amalekites symbolize the flesh. Like Amalek, the flesh is our sworn enemy. As a human being, the flesh is always there making war against the Spirit. The Spirit displaces the flesh from its territory much like Israel displaced Amalek. So the flesh is constantly trying to gain back that territory.