  We consider translators to the extrinsic flows in $\mathbb R^n\times\mathbb
R$ and $\mathbb H^n\times\mathbb R$ (called $r$-mean curvature flows or
$r$-MCF, for short) whose velocity functions are the higher order mean
curvatures $H_r.$ We show that there exist rotational bowl-type and
catenoid-type translators to $r$-MCF in both $\mathbb R^n\times\mathbb R$ and
$\mathbb H^n\times\mathbb R,$ and also that there exist parabolic and
hyperbolic catenoid-type translators to $r$-MCF in $\mathbb H^n\times\mathbb
R.$ In addition, we show that there exist Grim Reaper-type translators to
Gaussian flow ($n$-MCF) in $\mathbb R^n\times\mathbb R$ and $\mathbb
H^n\times\mathbb R$. We also establish the uniqueness of all these translators
(together with certain cylinders) among those which are invariant by either
rotations or translations (Euclidean, parabolic or hyperbolic). We apply this
uniqueness result to classify the translators to $r$-MCF in $\mathbb
R^n\times\mathbb R$ and $\mathbb H^n\times\mathbb R$ whose $r$-th mean
curvature is constant, as well as those which are isoparametric. Our results
extend to the context of $r$-MCF in $\mathbb R^n\times\mathbb R$ and $\mathbb
H^n\times\mathbb R$ the existence and uniqueness theorems by Altschuler--Wu (of
the bowl soliton) and Clutterbuck--Schn\"urer--Schulze (of the translating
catenoids) in Euclidean space.
