  Let $G$ and $H$ be hypergraphs on $n$ vertices, and suppose $H$ has large
enough minimum degree to necessarily contain a copy of $G$ as a subgraph. We
give a general method to randomly embed $G$ into $H$ with good "spread". More
precisely, for a wide class of $G$, we find a randomised embedding $f\colon
G\hookrightarrow H$ with the following property: for every $s$, for any partial
embedding $f'$ of $s$ vertices of $G$ into $H$, the probability that $f$
extends $f'$ is at most $O(1/n)^s$. This is a common generalisation of several
streams of research surrounding the classical Dirac-type problem. For example,
setting $s=n$, we obtain an asymptotically tight lower bound on the number of
embeddings of $G$ into $H$. This recovers and extends recent results of Glock,
Gould, Joos, K\"uhn, and Osthus and of Montgomery and Pavez-Sign\'e regarding
enumerating Hamilton cycles in Dirac hypergraphs. Moreover, using the recent
developments surrounding the Kahn--Kalai conjecture, this result implies that
many Dirac-type results hold robustly, meaning $G$ still embeds into $H$ after
a random sparsification of its edge set. This allows us to recover a recent
result of Kang, Kelly, K\"uhn, Osthus, and Pfenninger and of Pham, Sah,
Sawhney, and Simkin for perfect matchings, and obtain novel results for
Hamilton cycles and factors in Dirac hypergraphs. Notably, our randomised
embedding algorithm is self-contained and does not require Szemer\'edi's
regularity lemma or iterative absorption.
