  We investigate the effect of dephasing/decoherence on quantum transport
through open chaotic ballistic conductors in the semiclassical limit of small
Fermi wavelength to system size ratio, $\lambda_F/L << 1$. We use the
trajectory-based semiclassical theory to study a two-terminal chaotic dot with
decoherence originating from:
  (i) an external closed quantum chaotic environment,
  (ii) a classical source of noise,
  (iii) a voltage probe, i.e. an additional current-conserving terminal.
  We focus on the pure dephasing regime, where the coupling to the external
source of dephasing is so weak that it does not induce energy relaxation. In
addition to the universal algebraic suppression of weak localization, we find
an exponential suppression of weak-localization $\propto
\exp[-\tilde{\tau}/\tau_\phi]$, with the dephasing rate $\tau_\phi^{-1}$. The
parameter $\tilde{\tau}$ depends strongly on the source of dephasing. For a
voltage probe, $\tilde{\tau}$ is of order the Ehrenfest time $\propto \ln
[L/\lambda_F ]$. In contrast, for a chaotic environment or a classical source
of noise, it has the correlation length $\xi$ of the coupling/noise potential
replacing the Fermi wavelength $\lambda_F $. We explicitly show that the Fano
factor for shot noise is unaffected by decoherence. We connect these results to
earlier works on dephasing due to electron-electron interactions, and
numerically confirm our findings.
