  Rubidium Rydberg atoms in either $|m_j|$-sublevel of the $36p_{3/2}$ state
can exchange energy via Stark-tuned F\"{o}rster resonances, including two-,
three-, and four-body dipole-dipole interactions. Three-body interactions of
this type were first reported and categorized by Faoro, \textit{et al.}~[Nat.\
Commun.\ \textbf{6}, 8173 (2015)] and their Borromean nature was confirmed by
Tretyakov, \textit{et al.}~[Phys.\ Rev.\ Lett. \textbf{119}, 173402 (2017)]. We
report the time dependence of the $N$-body F\"{o}rster resonance $N\times
36p_{3/2,|m_j|=1/2}\rightarrow 36s_{1/2}+37s_{1/2}+(N-2)\times
36p_{3/2,|m_j|=3/2}$, for $N=2,3$, and 4, by measuring the fraction of
initially excited atoms that end up in the $37s_{1/2}$ state as a function of
time. The essential features of these interactions are captured in an
analytical model that includes only the many-body matrix elements and
neighboring atom distribution. A more sophisticated simulation reveals the
importance of beyond-nearest-neighbor interactions and of always-resonant
interactions.
