Exclusive processes probe the internal structure of nucleons through measurement of structure functions, which are sensitive to the chiral-even generalized parton distributions (GPDs) and the chiral-odd transversity GPDs. The GPDs encode correlations of parton distributions in longitudinal momentum space and in transverse impact parameter space through measurements of the $x_B$ and $t$ dependence of photon or meson production. Deeply virtual Compton scattering (DVCS) can access GPDs most directly through measurement of polarization observables in $ep\to ep\gamma$, while transversity GPDs may be accessed through exclusive pion production, e.g. $ep\to ep\pi^0$. The CW electron accelerator at Jefferson Lab with its beam energy of up to 6 GeV, in combination with the CLAS detector in Hall B, allowed measurements of these processes in large parts of the kinematical space in $Q^2$, $x_B$, and $t$. Measurements of both processes were carried out by scattering longitudinally polarized electrons from a dynamically polarized NH$_3$ target, where the spins of the free protons were aligned along the beam axis. Preliminary results will be presented for single and double polarization asymmetries of both processes, and compared to theoretical calculations using the handbag mechanism. Extensions of this experimental program to beam energies up to 12GeV with the upgraded CLAS12 detector will also be briefly discussed.