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Deep learning models have shown great promise in estimating tissue microstructure from limited diffusion magnetic resonance imaging data. However, these models face domain shift challenges when test and train data are from different scanners and protocols, or when the models are applied to data with inherent variations such as the developing brains of infants and children scanned at various ages. Several techniques have been proposed to address some of these challenges, such as data harmonization or domain adaptation in the adult brain. However, those techniques remain unexplored for the estimation of fiber orientation distribution functions in the rapidly developing brains of infants. In this work, we extensively investigate the age effect and domain shift within and across two different cohorts of 201 newborns and 165 babies using the Method of Moments and fine-tuning strategies. Our results show that reduced variations in the microstructural development of babies in comparison to newborns directly impact the deep learning models' cross-age performance. We also demonstrate that a small number of target domain samples can significantly mitigate domain shift problems.