How to punctuate multiline equations?

Mathematics are just symbolic sentences. Punctuate them as you would verbal ones. You can find some resources here. Read your maths aloud. If they sound ungrammatical you need to rewrite them. Your first example would read better as:

If we call t_c the damping constant, we can reduce f to the simple form f(t) = \exp{-t_c/t}.

And the punctuation would automatically make sense.

Although this may be negligible you might wish to consider removing the horizontal spacing added through proper punctuation within a display equation (by using \rlap). For example, you could use

The function \(g\) is simply the derivative of \(f\), so
\begin{align*}
  g(t) &= \deriv{f(t)}{t}\\
       &= \frac{t_c}{t^2}\exp{-t_c/t}\rlap{,}
\end{align*}
so \(g(t)\) will be smaller than \(f(t)\) for large \(t\).

to make sure that the widest stretch of align* is centered within the text block.