Scientific institutions play a crucial role in driving intellectual, social, and technological progress. Their capacity to innovate depends mainly on their ability to attract, retain, and nurture scientific talent and ultimately make it available to other organizations, industries, or the economy. As researchers change institutions during their careers, their skills are also transferred. The extent and mechanisms by which academic institutions manage their internal portfolio of scientific skills by attracting and sending researchers are far from being understood. We examine 25 million publication histories of 9.2 million scientists extracted from a large-scale bibliographic database covering thousands of research institutions worldwide to understand how the skills of mobile scientists align with those present in-house. We find a clear association between top-ranked institutions and greater skill alignment, i.e., the degree to which skills of incoming academics match those of their colleagues at the institution. We uncover similar high-alignment for scientists leaving top-ranked institutions. This type of academic alignment is more pronounced in engineering and life, health, earth, and physical sciences than in mathematics, computer science, social sciences, and the humanities. We show that over the past two decades, institutions generally have become more closely aligned in their overall skill profiles. We interpret these results in terms of levels of proactive management of the composition of the scientific workforce, diversity, and internal collaboration strategies at the institutional level.