  We investigate the critical current density and vortex dynamics in single
crystals of Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$ ($x$ = 0.051, 0.056, 0.065, and
0.073). The samples exhibit different critical temperatures and superconducting
phase fractions. We show that in contrast to their Ba-based counterpart, the
crystals do not exhibit a second peak in the field dependence of magnetization.
The calculated composition-dependent critical current density ($j_{\rm c}$)
increases initially with Co doping, maximizing at $x$ = 0.065, and then
decreases. This variation in $j_{\rm c}$ follows the superconducting phase
fractions in this series. The calculated $j_{\rm c}$ shows strong temperature
dependence, decreasing rapidly upon heating. Magnetic relaxation measurements
imply a nonlogarithmic dependence on time. We find that the relaxation rate is
large, reflecting weak characteristic pinning energy. The analysis of
temperature- and field-dependent magnetic relaxation data suggests that vortex
dynamics in these compounds is consistent with plastic creeping rather than the
collective creep model, unlike other 122 pnictide superconductors. This
difference may cause the absence of the second peak in the field dependent
magnetization of Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$.
