I posted several weeks ago about an Astra Model 4000 Tri-Caliber Kit that was offered in 1956-57. It came with a complete additional slide assembly and magazine in .22 LR and a .32 ACP and .380 ACP barrel for the regular slide assembly. Both of the centerfire magazines are identical and not marked as to caliber. Antaris' Astra book shows a photo of how it was packaged originally, and mine is the same.
I have not yet test fired it but it appears that all that is needed to swap between .32 and .380 is to swap barrels. Both magazines feed .32 A-Zoom dummy rounds into the .32 barrel when worked by hand and both mags feed dummy .380 rounds into the .380 barrel the same way. I assume that a change in recoil springs and even magazines is not necessary and the extractor must be designed in such a way as to work properly with both centerfire rounds. Perhaps the OP's Browning is just as easily changed, with just the barrel swap needed ?