\documentclass[beamer]{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing}
\usetikzlibrary{external}
\tikzexternalize % activate! 
\begin{document}
\begin{standaloneframe}

  % If overlays do not work, use \only<1-n>{...} where n is the max overlay
  \only<1-5>{
    \begin{tikzpicture}[auto]
      \node<1->[color=white] (whitestep1) at (0,6) {$T$ and $f:\mathcal S_n(T)\to k$};
      \node<1->[color=white] (whitestep2) at (0,4) {$S$ and $f:\mathcal S_n(S)\to k$ stable};
      \node<1->[color=white] (whitestep3) at (0,2) {$S$ and $f_{\mathrm{lim}}:\mathcal S_{n-1}(S)\to k$};
      \node<1->[color=white] (whitestep4) at (0,0) {$S'$ homogeneous for $f$};

%      \draw[color=white, decorate,decoration={snake, amplitude=.4mm}, ->] (whitestep1) -- (whitestep2) node[left, outer sep=0.3cm, midway, text width=2.5cm, align=right] {\scriptsize iteration of pigeonhole principle};
      \draw[color=white, decorate,decoration={snake, amplitude=.4mm}, ->] (whitestep1) -- (whitestep2) node[left, outer sep=0.3cm, midway, text width=2.5cm, align=right, color=white] {\scriptsize iteration of pigeonhole principle};
 %     \draw[decorate,decoration={snake, amplitude=.4mm}, ->] (whitestep1) -- (whitestep2);
      \draw[color=white, decorate,decoration={snake, amplitude=.4mm}, ->] (whitestep2) -- (whitestep3) node[left, outer sep=0.3cm, midway, align=right, color=white] {\scriptsize yields};
%      \draw[decorate,decoration={snake, amplitude=.4mm}, ->] (whitestep2) -- (whitestep3);
      \draw[color=white, decorate,decoration={snake, amplitude=.4mm}, ->] (whitestep3) -- (whitestep4) node[left, outer sep=0.3cm, midway, text width=2.5cm, align=right, color=white] {\scriptsize application of $\mathrm{MTT}^{n-1}$};
 %     \draw[decorate,decoration={snake, amplitude=.4mm}, ->] (whitestep3) -- (whitestep4);


      \node<1-> (step1) at (0,6) {$T$ and $f:\mathcal S_n(T)\to k$};
      \node<2-> (step2) at (0,4) {$S$ and $f:\mathcal S_n(S)\to k$ stable};
      \node<3-> (step3) at (0,2) {$S$ and $f_{\mathrm{lim}}:\mathcal S_{n-1}(S)\to k$};
      \node<4-> (step4) at (0,0) {$S'$ homogeneous for $f$};

%      \draw<6->[color=white, decorate,decoration={snake, amplitude=.4mm}, ->] (step1) -- (step2) node[left, outer sep=0.3cm, midway, text width=2.5cm, align=right] {\scriptsize iteration of pigeonhole principle};
      \draw<5->[color=white, decorate,decoration={snake, amplitude=.4mm}, ->] (step1) -- (step2) node[left, outer sep=0.3cm, midway, text width=2.5cm, align=right, color=black] {\scriptsize iteration of pigeonhole principle};
      \draw<2->[decorate,decoration={snake, amplitude=.4mm}, ->] (step1) -- (step2);
      \draw<3->[color=white, decorate,decoration={snake, amplitude=.4mm}, ->] (step2) -- (step3) node[left, outer sep=0.3cm, midway, align=right, color=black] {\scriptsize yields};
      \draw<3->[decorate,decoration={snake, amplitude=.4mm}, ->] (step2) -- (step3);
      \draw<4->[color=white, decorate,decoration={snake, amplitude=.4mm}, ->] (step3) -- (step4) node[left, outer sep=0.3cm, midway, text width=2.5cm, align=right, color=black] {\scriptsize application of $\mathrm{MTT}^{n-1}$};
      \draw<4->[decorate,decoration={snake, amplitude=.4mm}, ->] (step3) -- (step4);
    \end{tikzpicture}
  }
\end{standaloneframe}
\end{document}