I would like to animate the following picture so that the length of the blue box becomes smaller and smaller. Using the overprint environment along with \transduration{0.2} makes the job. However, this requires to copy and paste the piece of code, each time reducing the x-coordinate of topRight_corner. I am sure there is a proper way to do that...
PS: I am using this to illustrate the limit as 'delta t' goes to 0.
Here is the TikZ picture

and here is the piece of code I wrote for that
\documentclass{beamer}
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
%------ tikZ ------%
\usepackage{tikz}
\usetikzlibrary{positioning, arrows}
%------------------%
\begin{document}
\begin{frame}[c]
\begin{center}
\begin{tikzpicture}
\node[coordinate] at (0, 0) (bottomeLeft_corner) {};
\node[coordinate] at (2.50, 1.25) (topRight_corner) {};
\path[fill=blue!20] (bottomeLeft_corner) rectangle (topRight_corner);
\node[above left=0.625cm and 2cm of bottomeLeft_corner, coordinate] (start1) {};
\node[above left=0.625cm and 0cm of bottomeLeft_corner, coordinate] (end1) {};
\draw[semithick] (start1) -- (end1);
\node[above right=-0.625cm and 0cm of topRight_corner, coordinate] (start2) {};
\node[above right=0.625cm and 4.5cm of bottomeLeft_corner, coordinate] (end2) {};
\draw[->, >=stealth', semithick] (start2) -- (end2);
\draw[semithick, black!40] (end1) -- (start2);
\node[below=0cm of end2, xshift=0.25cm] {time};
\node[yshift=-0.25cm] at (bottomeLeft_corner) {$t$};
\end{tikzpicture}
\end{center}
\end{frame}
\end{document}


