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No, not every matroid satisfies this property. For example, it is known to fail for the cycle matroid of $K_4$. The matroids that satisfies your property are called base orderable matroids. There are important classes of matroids that are base orderable, such as transversal matroids. Moreover, base orderability is a minor-closed property, and Ingleton proved that there are actually an infinite number of excluded minors. See these slides of Joseph Bonin for more information. For example, the slides include a proof that $K_4$ is not base orderable.