  If dark matter annihilates into standard model particles with a cross-section
which is velocity dependent, then Local Group dwarf galaxies will not be the
best place to search for the resulting gamma ray emission. A greater flux would
be produced by more distant and massive halos, with larger velocity
dispersions. We construct full-sky predictions for the gamma-ray emission from
galaxy- and cluster-mass halos within $\sim 200 \, {\mathrm{Mpc}}$ using a
suite of constrained $N$-body simulations (CSiBORG) based on the Bayesian
Origin Reconstruction from Galaxies algorithm. Comparing to observations from
the Fermi Large Area Telescope and marginalising over reconstruction
uncertainties and other astrophysical contributions to the flux, we obtain
constraints on the cross-section which are two (seven) orders of magnitude
tighter than those obtained from dwarf spheroidals for $p$-wave ($d$-wave)
annihilation. We find no evidence for either type of annihilation from dark
matter particles with masses in the range $m_\chi = 2-500 \,
{\mathrm{GeV}}/c^2$, for any channel. As an example, for annihilations
producing bottom quarks with $m_\chi = 10 \, {\mathrm{GeV}}/c^2$, we find
$a_{1} < 2.4 \times 10^{-21} \, {\mathrm{cm^3 s^{-1}}}$ and $a_{2} < 3.0 \times
10^{-18} \, {\mathrm{cm^3 s^{-1}}}$ at 95% confidence, where the product of the
cross-section, $\sigma$, and relative particle velocity, $v$, is given by
$\sigma v = a_\ell (v/c)^{2\ell}$ and $\ell=1, 2$ for $p$-, $d$-wave
annihilation, respectively. Our bounds, although failing to exclude the thermal
relic cross-section for velocity-dependent annihilation channels, are among the
tightest to date.
