The accuracy of modern automatic speaker verification (ASV) systems, when trained exclusively on adult data, drops substantially when applied to children's speech. The scarcity of children's speech corpora hinders fine-tuning ASV systems for children's speech. Hence, there is a timely need to explore more effective ways of reusing adults' speech data. One promising approach is to align vocal-tract parameters between adults and children through children-specific data augmentation, referred here to as ChildAugment. Specifically, we modify the formant frequencies and formant bandwidths of adult speech to emulate children's speech. The modified spectra are used to train ECAPA-TDNN (emphasized channel attention, propagation, and aggregation in time-delay neural network) recognizer for children. We compare ChildAugment against various state-of-the-art data augmentation techniques for children's ASV. We also extensively compare different scoring methods, including cosine scoring, PLDA (probabilistic linear discriminant analysis), and NPLDA (neural PLDA). We also propose a low-complexity weighted cosine score for extremely low-resource children ASV. Our findings on the CSLU kids corpus indicate that ChildAugment holds promise as a simple, acoustics-motivated approach, for improving state-of-the-art deep learning based ASV for children. We achieve up to 12.45% (boys) and 11.96% (girls) relative improvement over the baseline.