The algebra of Dirichlet series A(C+) consists on
those Dirichlet series convergent in the right half-plane C+ and
which are also uniformly continuous there. This algebra was recently introduced
by Aron, Bayart, Gauthier, Maestre, and Nestoridis. We describe the symbols
Φ:C+→C+ giving rise to bounded composition
operators CΦ in A(C+) and denote this
class by GA. We also characterise when the operator
CΦ is compact in A(C+). As a
byproduct, we show that the weak compactness is equivalent to the compactness
for CΦ. Next, the closure under the local uniform convergence
of several classes of symbols of composition operators in Banach spaces of
Dirichlet series is discussed. We also establish a one-to-one correspondence
between continuous semigroups of analytic functions {Φt} in the class
GA and strongly continuous semigroups of composition
operators {Tt}, Ttf=f∘Φt,
f∈A(C+). We conclude providing examples showing the
differences between the symbols of bounded composition operators in
A(C+) and the Hardy spaces of Dirichlet series
Hp and H∞