The psychological side of this same approach is the James-Lange theory, and it gives a better sense of what we actually observe. The theory is that emotions arise substantially before they are felt, and that the experience of feeling them occurs mostly at the point when you have decided what they mean.
The most obvious part of the emotional classification that can make it very different is whether it gets labeled initially as negative or positive. So the same physiological reaction can be fear or glee, anxiety or anticipation, anger or passionate approval, sorrow or exceptionally deep joy.
There are a lot of other dimensions, like the emotional alchemy that turns fear into anger or anger into sorrow as it arises. But just the fact that the significance of feelings that we admit really feel very much the same can be very different should be enough to convince you that the process itself is taking place. Observations about the process after this point usually come out of psychotherapy rather than neurology, so they are less convincing -- but I am going to openly embrace them anyway.
Once we feel the feeling, we then evolve it according to how past feelings have played out for us. The most common example here is the 'flight, flee, freeze or merge' response to fear. Sudden and intense fear can choose very different paths, to increase our energy so that we can't restrain ourselves and we have to choose to use it for either defense or escape, or it can create severe awareness of the source and lead us to try to be very far from it or to be very close to it. The choice feels like it is being made for us by our bodies. But it is really a decision we are making about history and interpretation.
The same kind of alchemy that happens as an emotion arises, can continue taking place as it evolves. When they are aware of them at all, people often describe these as emotional reactions to the original emotion, but that is a mental framing after the fact, we do not detect two separate physiological events, just a single ongoing process.
If we need to control our expression of the emotion, once it has been felt and elaborated, we often need to draw out the elaboration and change it further. Men, in particular, may modify emotions in elaborate ways that obscure them completely. Someone may often feel anger rise when the obvious reaction that the stimulus ought to provoke is fear and then, if they are powerless to really accomplish anything with the energy that would release, they turn that anger into sorrow and use the acquired skills of their gender role to suppress it. Or sorrow may arise and become fear because it represents a weakness that was not foreseen, and the empathic outreach that should occur needs to be frozen so that it does not flow out to a target that would not be safe to empathize with.
So yes. Emotions change as they happen. And expressing them is one of the things that might change them, since the expression is a social negotiation with important consequences.