Simulations of the two-dimensional Ising and three-state Potts models at their critical points are performed using the invaded cluster (IC) algorithm. It is argued that observables measured on a sublattice of size l should exhibit a crossover to Swendsen-Wang (SW) behavior for l sufficiently less than the lattice size L, and a scaling form is proposed to describe the crossover phenomenon. It is found that the energy autocorrelation time ${\ensuremath{\tau}}_{\ensuremath{\varepsilon}}(l,L)$ for an $l\ifmmode\times\else\texttimes\fi{}l$ sublattice attains a maximum in the crossover region, and a dynamic exponent
zIC for the IC algorithm is defined according to ${\ensuremath{\tau}}_{\ensuremath{\varepsilon},\mathrm{max}}\ensuremath{\sim}{L}^{{z}^{\mathrm{IC}}}.$ Simulation results for the three-state model yield ${z}^{\mathrm{IC}}=0.346\ifmmode\pm\else\textpm\fi{}0.002,$ which is smaller than values of the dynamic exponent found for the SW and Wolff algorithms and also less than the Li-Sokal bound. The results are less conclusive for the Ising model, but it appears that
zIC<0.21 and possibly that ${\ensuremath{\tau}}_{\ensuremath{\varepsilon},\mathrm{max}}\ensuremath{\sim}\mathrm{ln}L$ so that
zIC=0---similar to previous results for the SW and Wolff algorithms.