  Superrotations of asymptotically flat spacetimes in four dimensions can be
interpreted in terms of including cosmic strings within the phase space of
allowed solutions. In this paper we explore the implications of the inclusion
of cosmic branes on the asymptotic structure of vacuum spacetimes in dimension
d > 4. We first show that only cosmic (d-3)-branes are Riemann flat in the
neighbourhood of the brane, and therefore only branes of such dimension passing
through the celestial sphere can respect asymptotic local flatness. We derive
the asymptotically locally flat boundary conditions associated with including
cosmic branes in the phase space of solutions. We find the asymptotic expansion
of vacuum spacetimes in d=5 with such boundary conditions; the expansion is
polyhomogenous, with logarithmic terms arising at subleading orders in the
expansion. The asymptotically locally flat boundary conditions identified here
are associated with an extended asymptotic symmetry group, which may be
relevant to soft scattering theorems and memory effects.
