ltablex combines longtable and tabularx. longtable provides commands to define a first and a general header as well as a general and a last footer. These commands are actually used in the (admittedly brief) ltablex documentation, but a continued caption, which would be used in the general header (or footer), is not part of the example. The header/footer commands are described in the longtable documentation.
The code given below shows their use with ltablex. The continued caption is defined in the general header. I've also added an example using longtabu from the more recent tabu package, which offers a convenient way to define flexible X columns.

\documentclass{article}
\usepackage{lipsum}
\usepackage{booktabs}
\usepackage{ltablex}
%if longtabu is used, longtable needs to be loaded (which ltablex already does in this document)
%\usepackage{longtable}
\usepackage{tabu}
\textheight4in %to force a page break after a few rows
\renewcommand{\arraystretch}{1.5} %improve spacing
\begin{document}
\begin{tabularx}{\textwidth}{XX}
\caption{A Table}\\
\toprule
First Header & \\
\toprule
\endfirsthead
\caption{A Table (continued)}\\
\toprule
Header & \\
\toprule
\endhead
\bottomrule
Footer & \\
\bottomrule
\endfoot
\bottomrule
Last Footer & \\
\bottomrule
\endlastfoot
Text & \lipsum[75]\\
Text & \lipsum[75]\\
Text & \lipsum[75]\\
\end{tabularx}
\begin{longtabu} to \textwidth {X[1,l] %aligned left
X[2,l]} %twice as wide as first column
\caption{A Table}\\
\toprule
First Header & \\
\toprule
\endfirsthead
\caption{A Table (continued)}\\
\toprule
Header & \\
\toprule
\endhead
\bottomrule
Footer & \\
\bottomrule
\endfoot
\bottomrule
Last Footer & \\
\bottomrule
\endlastfoot
Text & \lipsum[75]\\
Text & \lipsum[75]\\
Text & \lipsum[75]\\
\end{longtabu}
\end{document}