Equal size underbraces at a lower position

You use a \parbox-based approach along the following lines:

\documentclass{article}
\usepackage{amsmath}
\usepackage{newtxtext,newtxmath} % optional - Times Roman text and math fonts
\newlength\mylen
\settowidth\mylen{$\displaystyle\alpha\circ X_{t-1}$} % measure width of widest element
\newcommand\mybox[1]{\parbox{\mylen}{\centering$\displaystyle #1$}}
\begin{document}
\[
\underbrace{\mybox{X_t}}_{\text{Population at time $t$}} =
\underbrace{\mybox{\alpha\circ X_{t-1}}}_{\text{Survivors from time $t-1$}} +
\underbrace{\mybox{P\epsilon_t}}_{\text{Immigration}}
\]
\end{document}

One very ugly way to do it is to cheat with some \vphantom. Please note that you can also insert horizontal spaces with \hspace* instead of a series of backslashes. There is probably a better way to achieve all this but, for a start...

\documentclass{article}
\usepackage{amsmath}

\begin{document}
$$
\underbrace{\vphantom{\Big|} \hspace*{5mm} X_t \hspace*{5mm}}_\text{Population at time $t$} =
\underbrace{\vphantom{\Big|}\alpha \circ X_{t-1}}_\text{Survivors from time $t-1$} +
\underbrace{\vphantom{\Big|} \hspace*{5mm} \epsilon_t \hspace*{5mm}}_\text{Immigration} 
$$

\end{document}

You can define \ubrace macro and use \def\ubracew{max-width-formula} before each such formula inside math mode.

\def\ubrace#1#2{%
   \setbox0=\hbox{$\displaystyle\ubracew$}%
   \underbrace{\hbox to\wd0{\hss$\displaystyle#1$\lower.8ex\hbox{}\hss}}%
   _{\text{#2}}%
}

$$
\def\ubracew{\alpha\circ X_{t-1}}
\ubrace{X_t}{Population at time $t$} =
\ubrace{\alpha \circ X_{t-1}}{Survivors from time $t-1$} +
\ubrace{\epsilon_t}{Immigration}
$$