  Dynamic Complexity studies the maintainability of queries with logical
formulas in a setting where the underlying structure or database changes over
time. Most often, these formulas are from first-order logic, giving rise to the
dynamic complexity class DynFO. This paper investigates extensions of DynFO in
the spirit of parameterised algorithms. In this setting structures come with a
parameter $k$ and the extensions allow additional "space" of size $f(k)$ (in
the form of an additional structure of this size) or additional time $f(k)$ (in
the form of iterations of formulas) or both. The resulting classes are compared
with their non-dynamic counterparts and other classes. The main part of the
paper explores the applicability of methods for parameterised algorithms to
this setting through case studies for various well-known parameterised
problems.
