The muon-spin-relaxation (\ensuremath{\mu}SR) technique has been used to study the internal field modulation dynamics of fluxons in crystals of \ensuremath{\kappa}-[BEDT-TTF${]}_{2}$Cu[NCS${]}_{2}$ (${\mathit{T}}_{\mathit{c}}$\ensuremath{\approxeq}10.5 K). The data exhibit a strong field dependence for applied fields in the range 129 Oe\ensuremath{\le}\ensuremath{\Vert}${\mathbf{H}}_{\mathrm{ext}}$\ensuremath{\Vert}\ensuremath{\le}4 kOe, which is attributed to a combination of vortex motion, finite core size effects, and pinning. In low fields (\ensuremath{\Vert}${\mathbf{H}}_{\mathrm{ext}}$\ensuremath{\Vert}=129 Oe), the data exhibit a flux-pinning transition at ${\mathit{T}}_{\mathit{x}}$\ensuremath{\sim}5 K, as evidenced by a peak in the second moment of the local field distribution at \ensuremath{\sim}5 K accompanied by an enhanced minimum in the first moment at the same temperature. In higher fields (\ensuremath{\Vert}${\mathbf{H}}_{\mathrm{ext}}$\ensuremath{\Vert}=750 Oe, 3 kOe, and 4 kOe), where the effects of motion and pinning are minimized, the symmetry of the underlying gap function is confirmed as s wave. Corrections for finite fluxon core, finite mean free path, and flux-lattice disorder were applied in determining the London penetration depth, ${\ensuremath{\lambda}}_{\mathit{L}}^{(\mathit{b}\mathit{c})}$(0)=7680\ifmmode\pm\else\textpm\fi{}700 \AA{}.