  Distillation of entanglement using only Gaussian operations is an important
primitive in quantum communication, quantum repeater architectures, and
distributed quantum computing. Existing distillation protocols for continuous
degrees of freedom are only known to converge to a Gaussian state when
measurements yield precisely the vacuum outcome. In sharp contrast,
non-Gaussian states can be deterministically converted into Gaussian states
while preserving their second moments, albeit by usually reducing their degree
of entanglement. In this work - based on a novel instance of a non-commutative
central limit theorem - we introduce a picture general enough to encompass the
known protocols leading to Gaussian states, and new classes of protocols
including multipartite distillation. This gives the experimental option of
balancing the merits of success probability against entanglement produced.
