Can I automatically break a long `displayquote` within `figure` across multiple pages?

Question

I have a figure environment containing a displayquote environment. The text is too long to fit on a single page, even if I change the text size (which I have been able to do elsewhere).

However, the displayquote does not break at the bottom of the page and instead just runs off it. See:

There are some similar questions (e.g., this one), which involve figures made up of subfigures, but I only have a single displayquote so these aren't applicable here.

So far, the best I've been able to do is to split it across multiple figures, with \clearpages between them.

This feels like a bit of a hack though, and results in the final chunk of the text being vertically-centered, even with [t!] on its figure (although this isn't happening with my MWE so I'm not sure where the behaviour is coming from).

Is there any way I can just have a single figure, with a single displayquote, that will automatically split itself across multiple pages? Even better, can I do so whilst automatically appending some sort of ‘continued on following page’ note to the end of each page?

MWE:

\documentclass[a4paper,12pt]{report}
\usepackage{blindtext}

\usepackage{caption} 
\usepackage[style=british]{csquotes}
\begin{document}
% Issue

\begin{figure}
    \begin{displayquote}
        \Blindtext
        \Blindtext
    \end{displayquote}
    \caption{Caption}
\end{figure}

%% Hacky fix

\begin{figure}
    \begin{displayquote}
        \blindtext

        \blindtext

        \blindtext
    \end{displayquote}
\end{figure}

\clearpage

\begin{figure}
    \begin{displayquote}
        \blindtext
    \end{displayquote}
    \caption{Lorem ipsum dolor sit amet}
    \label{fig:testlabel}
\end{figure}

Hey look at fig.~\ref{fig:testlabel}. % Should be `fig. 1'


\end{document}