We will consider a nonstationary sequence of independent random elements of a compact metrizable group. Assuming a natural nondegeneracy condition we will establish a weak-* convergence to the Haar measure, Ergodic Theorem, and Large Deviation Type Estimate. In particular, we will prove a nonstationary analog of classical Ito-Kawada theorem and give a new alternative proof for the stationary case. We will also show that all the results can be carried over to the case of a composition of random isometries.