Find generating function using Dynkin's formula : How is it possible since $f(x)=e^{xu}$ is not compact supported?

Question

(It looks long, but it's not). The aim of the question is to understand why can we apply Dynkins formula to $f(x)=e^{ux}$ whereas $f$ is no compact supported ?

Here a lemma and Dynkin's formula and a lemma :

Here the application

Question

1) First, at the red cross, I don't understand $f(X(t),t)=f(x)=e^{ux}$. It doesn't make any sense, does it ?

2) Then, how can we apply Dynkin's formula to $f(x)=e^{ux}$ whereas ti's not compact supported ?