In this paper, we study "nonstationary consistency" of subspace methods for eigenstructure identification, i.e., the ability of subspace algorithms to converge to the true eigenstructure despite nonstationarities in the excitation and measurement noises. Note that such nonstationarities may result in having time-varying zeros for the underlying system, so the problem is nontrivial. In particular, likelihood- and prediction-error related methods do not ensure consistency under such situation, because estimation of poles and estimation of zeros are tightly coupled. We show in turn that subspace methods ensure such consistency. Our study carefully separates statistical from nonstatistical arguments, therefore, enlightening the role of statistical assumptions in this story.