Play animated graphic when "next slide" is clicked

Put

\alt<2>{%
  \animategraphics[scale=0.4,autoplay]{8}{movie}{000}{200}%
}{%
  \includegraphics[scale=0.4]{movie000}%
}

into the frame body.

Complete example using multipage PDF from pkg mwe as animation frames:

\documentclass{beamer}

\usepackage{animate}

\begin{document}

\begin{frame}[t]{Animation}
\dots started on ``next slide'':\\[2ex]

\alt<2>{%
  \animategraphics[height=2in,autoplay]{8}{example-image-a4-numbered}{}{}%
}{%
  \includegraphics[height=2in,page=1]{example-image-a4-numbered}%
}
\end{frame}

\end{document}

An easy way to avoid code duplication is described in this workaround:

• variable \myframe defines the number of frames composing the animation

• overlay command \only overwrites the value of \myframe depending on which (beamer) frame you are.

  \documentclass{beamer}
  \usepackage{default}
  \usepackage{tikz}
  \usepackage{animate}
  \usepackage{xcolor}
  
  \begin{document}
  \begin{frame}{autoplay next slide example}
  %% variable myframe decides how many frames the animation is composed of
  \only<1>{\def\myframes{1}}
  \only<2>{\def\myframes{81}}
  
  \begin{animateinline}[poster=first,autoplay]{12}%
  %%  variable rl defines the percentual variation in the position of the moving object
  \multiframe{\myframes}{rl=0+.0125}{%
  
      \begin{tikzpicture}
          \clip (-.5,-.5) rectangle (10,7); %% visible rectangle              
          \draw[draw=black, fill=black] (1+3*\rl, 2+4*\rl) circle (3pt);          
      \end{tikzpicture}      
  }
  \end{animateinline}     
  \end{frame}
  \end{document}