Total reaction cross sections ${\ensuremath{\sigma}}_{R}$ of $(30--60)A\mathrm{MeV}$ ${}^{4,6,8}\mathrm{He}$ and ${}^{6,7,8,9,11}\mathrm{Li}$ on Pb, and $2n$-removal cross sections ${\ensuremath{\sigma}}_{\ensuremath{-}2n}$ of ${}^{6,8}\mathrm{He}$ and ${}^{11}\mathrm{Li}$ on Pb, were measured by injecting magnetically separated, focused, monoenergetic, secondary beams of those projectiles into a telescope containing Pb targets separated by thin Si detectors. All these ${\ensuremath{\sigma}}_{R}$'s (except ${}^{4}\mathrm{He}),$ and ${\ensuremath{\sigma}}_{\ensuremath{-}2n}$ for ${}^{6}\mathrm{He}$ and ${}^{11}\mathrm{Li},$ are underpredicted by microscopic model calculations which include only nuclear forces. Better agreement is achieved by including electromagnetic dissociation in the model, for those projectiles for which either the electric dipole response functions or the dominant photodissociation cross sections were known. The cross sections ${\ensuremath{\sigma}}_{\ensuremath{-}4n}$ for ${}^{8}\mathrm{He},$ ${\ensuremath{\sigma}}_{\ensuremath{-}\mathrm{xn}}$ for ${}^{7,8,9}\mathrm{Li},$ and $({\ensuremath{\sigma}}_{\ensuremath{-}3n}+{\ensuremath{\sigma}}_{\ensuremath{-}4n})$ for ${}^{11}\mathrm{Li}$ were found to be \ensuremath{\leqslant}0.7 b. All ${\ensuremath{\sigma}}_{R}$'s were measured to better than 5% accuracy, showing that the method is usable for other target elements sandwiched into a Si telescope.