  Building on Quillen's rational homotopy theory, we obtain algebraic models
for the rational homotopy theory of parametrised spectra. For any
simply-connected space $X$ there is a dg Lie algebra $\Lambda_X$ and a
(coassociative cocommutative) dg coalgebra $C_X$ that model the rational
homotopy type. In this article, we prove that the rational homotopy type of an
$X$-parametrised spectrum is completely encoded by a $\Lambda_X$-representation
and also by a $C_X$-comodule. The correspondence between rational parametrised
spectra and algebraic data is obtained by means of symmetric monoidal
equivalences of homotopy categories that vary pseudofunctorially in the
simply-connected parameter space $X$.
  Our results establish a comprehensive dictionary enabling the translation of
topological constructions into homological algebra using Lie representations
and comodules, and conversely. For example, the fibrewise smash product of
parametrised spectra is encoded by the derived tensor product of dg Lie
representations and also by the derived cotensor product of dg comodules. As an
application, we obtain algebraic descriptions of rational homotopy classes of
fibrewise stable maps, providing new tools for the study of spaces of sections.
