Suppose that
(
X
,
G
)
(X,G)
is a second countable locally compact transformation group. We let
S
G
(
X
)
\operatorname {S}_G(X)
denote the set of Morita equivalence classes of separable dynamical systems
(
A
,
G
,
α
)
(A,G,\alpha )
where
A
A
is a
C
0
(
X
)
C_{0}(X)
-algebra and
α
\alpha
is compatible with the given
G
G
-action on
X
X
. We prove that
S
G
(
X
)
\operatorname {S}_{G}(X)
is a commutative semigroup with identity with respect to the binary operation
[
A
,
G
,
α
]
[
B
,
G
,
β
]
=
[
A
⊗
X
B
,
G
,
α
⊗
X
β
]
[A,G,\alpha ][B,G,\beta ]=[A\otimes _{X}B,G,\alpha \otimes _{X}\beta ]
for an appropriately defined balanced tensor product on
C
0
(
X
)
C_{0}(X)
-algebras. If
G
G
and
H
H
act freely and properly on the left and right of a space
X
X
, then we prove that
S
G
(
X
/
H
)
\operatorname {S}_{G}(X/H)
and
S
H
(
G
∖
X
)
\operatorname {S}_{H}(G\setminus X)
are isomorphic as semigroups. If the isomorphism maps the class of
(
A
,
G
,
α
)
(A,G,\alpha )
to the class of
(
B
,
H
,
β
)
(B,H,\beta )
, then
A
⋊
α
G
A\rtimes _{\alpha }G
is Morita equivalent to
B
⋊
β
H
B\rtimes _{\beta }H
.