Suppose we have some function $f$ which depends on a time parameter $t$ and another parameter $s$, suppose $f$ satisfies some Partial Delay Differential Equation: $$ \frac{\partial f(t,s)}{\partial t} + \frac{\partial f(t,s)}{\partial s} = F(f(t,s_1),\dots,f(t,s_n)). \qquad (1) $$ for some $s_1,\dots,s_n\leq s$. Then one can find the equilibirum, i.e. the behaviour of $f$ in the limit $t\rightarrow \infty$ by leaving out the dependence on $t$, which yields the delay differential equation: $$ f'(s)=F(f(s_1),\dots,f(s_n)), \qquad\qquad\qquad\qquad\qquad (2) $$ which is often easier to solve.
How does one go about proving that this is a valid strategy, I mean prove that the solution of (2) indeed corresponds to the limit of the solution of (1)?