Abstract Low dust opacity spectral indices (β < 1) measured in the inner envelopes of class 0/I young stellar objects (age ∼10^4–5 yr) have been interpreted as the presence of (sub-)millimeter dust grains in these environments. The density conditions and the lifetimes of collapsing envelopes have proven unfavorable for the growth of solids up to millimeter sizes. As an alternative, magnetohydrodynamical simulations suggest that protostellar jets and outflows might lift grains from circumstellar disks and diffuse them in the envelope. We reframe available data for the CALYPSO sample of Class 0/I sources and show tentative evidence for an anticorrelation between the value of β _{1–3 mm} measured in the inner envelope and the mass-loss rate of their jets and outflows, supporting a connection between the two. We discuss the implications that dust transport from the disk to the inner envelope might have for several aspects of planet formation. Finally, we urge for more accurate measurements of both correlated quantities and the extension of this work to larger samples, necessary to further test the transport scenario.