  Kinematic algebras can be realised on geometric spaces and constrain the
physical models that can live on these spaces. Different types of kinematic
algebras exist and we consider the interplay of these algebras for
non-relativistic limits of a relativistic system, including both the Galilei
and the Carroll limit. We develop a framework that captures systematically the
corrections to the strict non-relativistic limit by introducing new
infinite-dimensional algebras, with emphasis on the Carroll case. One of our
results is to highlight a new type of duality between Galilei and Carroll
limits that extends to corrections as well. We realise these algebras in terms
of particle models. Other applications include curvature corrections and
particles in a background electro-magnetic field.
