Let
π
:
X
→
X
0
\pi :X \rightarrow X_{0}
be a projective morphism of schemes, such that
X
0
X_{0}
is Noetherian and essentially of finite type over a field
K
K
. Let
i
∈
N
0
i \in \mathbb {N}_{0}
, let
F
{\mathcal {F}}
be a coherent sheaf of
O
X
{\mathcal {O}}_{X}
-modules and let
L
{\mathcal {L}}
be an ample invertible sheaf over
X
X
. Let
Z
0
⊆
X
0
Z_{0} \subseteq X_{0}
be a closed set. We show that the depth of the higher direct image sheaf
R
i
π
∗
(
L
n
⊗
O
X
F
)
{\mathcal {R}}^{i}\pi _{*}({\mathcal {L}}^{n} \otimes _{{\mathcal {O}}_{X}} {\mathcal {F}})
along
Z
0
Z_{0}
ultimately becomes constant as
n
n
tends to
−
∞
-\infty
, provided
X
0
X_{0}
has dimension
≤
2
\leq 2
. There are various examples which show that the mentioned asymptotic stability may fail if
dim
(
X
0
)
≥
3
\dim (X_{0}) \geq 3
. To prove our stability result, we show that for a finitely generated graded module
M
M
over a homogeneous Noetherian ring
R
=
⨁
n
≥
0
R
n
R=\bigoplus _{n \geq 0}R_{n}
for which
R
0
R_{0}
is essentially of finite type over a field and an ideal
a
0
⊆
R
0
\mathfrak {a}_{0} \subseteq R_{0}
, the
a
0
\mathfrak {a}_{0}
-depth of the
n
n
-th graded component
H
R
+
i
(
M
)
n
H^{i}_{R_{+}}(M)_{n}
of the
i
i
-th local cohomology module of
M
M
with respect to
R
+
:=
⨁
k
>
0
R
k
R_{+}:=\bigoplus _{k>0}R_{k}
ultimately becomes constant in codimension
≤
2
\leq 2
as
n
n
tends to
−
∞
-\infty
.