  When we are interested in the long-term behaviour of solutions to linear
evolution equations, a large variety of techniques from the theory of
$C_0$-semigroups is at our disposal. However, if we consider for instance
parabolic equations with unbounded coefficients on $\mathbb{R}^d$, the solution
semigroup will not be strongly continuous, in general. For such semigroups many
tools that can be used to investigate the asymptotic behaviour of
$C_0$-semigroups are not available anymore and, hence, much less is known about
their long-time behaviour.
  Motivated by this observation, we prove new characterisations of the operator
norm convergence of general semigroup representations - without any time
regularity assumptions - by adapting the concept of the "semigroup at
infinity", recently introduced by M.~Haase and the second named author. Besides
its independence of time regularity, our approach also allows us to treat the
discrete-time case (i.e., powers of a single operator) and even more abstract
semigroup representations within the same unified setting.
  As an application of our results, we prove a convergence theorem for
solutions to systems of parabolic equations with the aforementioned properties.
