In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space
R
n
+
1
\mathbb {R}^{n+1}
with speed
ψ
u
α
ρ
δ
f
−
β
\psi u^\alpha \rho ^\delta f^{-\beta }
, where
ψ
\psi
is a smooth positive function on unit sphere,
u
u
is the support function of the hypersurface,
ρ
\rho
is the radial function,
f
f
is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. When
ψ
=
1
\psi =1
, we prove that the flow exists for all time and converges to infinity if
α
+
δ
+
β
⩽
1
\alpha +\delta +\beta \leqslant 1
, and
α
⩽
0
>
β
\alpha \leqslant 0>\beta
, while in case
α
+
δ
+
β
>
1
\alpha +\delta +\beta >1
,
α
,
δ
⩽
0
>
β
\alpha ,\delta \leqslant 0>\beta
, the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to a sphere centered at the origin. In particular, the results of Gerhardt [J. Differential Geom. 32 (1990), pp. 299–314; Calc. Var. Partial Differential Equations 49 (2014), pp. 471–489] and Urbas [Math. Z. 205 (1990), pp. 355–372] can be recovered by putting
α
=
δ
=
0
\alpha =\delta =0
. Our previous works [Proc. Amer. Math. Soc. 148 (2020), pp. 5331–5341; J. Funct. Anal. 282 (2022), p. 38] and Hu, Mao, Tu and Wu [J. Korean Math. Soc. 57 (2020), pp. 1299–1322] can be recovered by putting
δ
=
0
\delta =0
and
α
=
0
\alpha =0
respectively. By the convergence of these flows, we can give a new proof of uniqueness theorems for solutions to
L
p
L^p
-Minkowski problem and
L
p
L^p
-Christoffel-Minkowski problem with constant prescribed data. Similarly, we consider the
L
p
L^p
dual Christoffel-Minkowski problem and prove a uniqueness theorem for solutions to
L
p
L^p
dual Minkowski problem and
L
p
L^p
dual Christoffel-Minkowski problem with constant prescribed data. At last, we focus on the long time existence and convergence of a class of anisotropic flows (i.e. for general function
ψ
\psi
). The final result not only gives a new proof of many previously known solutions to
L
p
L^p
dual Minkowski problem,
L
p
L^p
-Christoffel-Minkowski problem, etc. by such anisotropic flows, but also provides solutions to
L
p
L^p
dual Christoffel-Minkowski problem with some conditions.