We use the quasiparticle random-phase approximation with a few Skyrme density functionals to calculate strength functions in the J(pi)=0(+),1(-), and 2(+) channels for even Ca, Ni, and Sn isotopes from the proton drip line to the neutron drip line. We show where and how low-lying strength begins to appear as N increases. We also exhibit partial energy-weighted sums of the transition strength as functions of N for all nuclei calculated and transition densities for many of the interesting peaks. We find that low-energy strength increases with N in all multipoles, but with distinctive features in each. The low-lying 0(+) strength at large N barely involves protons at all, with the strength coming primarily from a single two-quasineutron configuration with very large spatial extent. The low-lying 1(-) strength is different, with protons contributing to the transition density in the nuclear interior together with neutrons at large radii and with somewhat more collectivity in the heavier isotopes. The low-lying 2(+) transition strength goes largely to more localized and frequently more collective states. In all channels the correlation between strength and collectivity is much weaker than in stable nuclei. The three Skyrme interactions we test produce similar results, differing most significantly in their predictions for the location of the neutron drip line, the boundaries of deformed regions, energies of and transition strengths to the lowest 2(+) states between closed shells, and isovector energy-weighted sum rules.