  We study asymptotic behaviour of positive ground state solutions of the
nonlinear Choquard equation $$ -\Delta u+\varepsilon u=(I_\alpha \ast
|u|^{p})|u|^{p-2}u+ |u|^{q-2}u \quad {\rm in} \ \mathbb R^N, $$ where $N\ge 3$
is an integer, $p\in [\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}]$, $q\in
(2,\frac{2N}{N-2}]$, $I_\alpha$ is the Riesz potential and $\varepsilon>0$ is a
parameter. We show that as $\varepsilon\to 0$ (resp. $\varepsilon\to \infty$),
after a suitable rescaling the ground state solutions of $(P_\varepsilon)$
converge in $H^1(\mathbb R^N)$ to a particular solution of some limit
equations. We also establish a sharp asymptotic characterisation of such a
rescaling, and the exact asymptotic behaviours of $u_\varepsilon(0), \|\nabla
u_\varepsilon\|_2^2, \|u_\varepsilon\|_2^2, \int_{\mathbb R^N}(I_\alpha\ast
|u_\varepsilon|^p)|u_\varepsilon|^p$ and $\|u_\varepsilon\|_q^q$, which depend
in a non-trivial way on the exponents $p, q$ and the space dimension $N$. We
also discuss a connection of our results with an associated mass constrained
problem with normalization constraint $\int_{\mathbb R^N}|u|^2=c^2$. As a
consequence of the main results, we obtain the existence, multiplicity and
exact asymptotic behaviour of positive normalized solutions of such a problem
as $c\to 0$ and $c\to \infty$.
