  Let $A$ and $B$ be $C^*$-algebras, $A$ separable and $I$ an ideal in $B$. We
show that for any completely positive contractive linear map $\psi\colon A\to
B/I$ there is a continuous family $\Theta_t\colon A\to B$, for $t\in
[1,\infty)$, of lifts of $\psi$ that are asymptotically linear, asymptotically
completely positive and asymptotically contractive. If $\psi$ is orthogonality
preserving, then $\Theta_t$ can be chosen to have this property asymptotically.
If $A$ and $B$ carry continuous actions of a second countable locally compact
group $G$ such that $I$ is $G$-invariant and $\psi$ is equivariant, we show
that the family $\Theta_t$ can be chosen to be asymptotically equivariant. If a
linear completely positive lift for $\psi$ exists, we can arrange that
$\Theta_t$ is linear and completely positive for all $t\in [1,\infty)$. In the
equivariant setting, if $A$, $B$ and $\psi$ are unital, we show that
asymptotically linear unital lifts are only guaranteed to exist if $G$ is
amenable. This leads to a new characterization of amenability in terms of the
existence of asymptotically equivariant unital sections for quotient maps.
