  We consider how the swampland criteria might be applied to models in which
scalar fields have nontrivial kinetic terms, particularly in the context of
$P(\phi,X)$ theories, popularly used in approaches to inflation, to its
alternatives, and to the problem of late-time cosmic acceleration. By embedding
such theories in canonical multi-field models, from which the original theory
emerges as a low-energy effective field theory, we derive swampland
constraints, and study the circumstances under which these might be evaded
while preserving cosmologically interesting phenomenology. We further
demonstrate how these successes are tied to the phenomenon of turning in field
space in the multi-field picture. We study both the general problem and
specific examples of particular interest, such as DBI inflation.
