I posted an article a few doors down on this site.
Here is part of it and the last paragraph pertains to your question.
It isn't 100% clear to me, but I believe they do provided they are of
the same sort.
243=308 <>fit the 358. I think they will.
All Browning BLR rifles produced between 1970 and 1980 are correctly referred to as Browning BLR’s. All Browning BLR Rifles produced between 1981and 1994 are correctly referred to as Browning BLR Model 81’s and could be purchased in either short action, or long action after 1991. All BLR Rifles produced from 1995 through 2002 are correctly referred to as the Browning BLR Lightning Model (Alloy receiver). All Browning BLR models produced after 2003 are correctly referred to as Browning BLR Model 81 Lightweights or BLR Lightweights.
Keep in mind that the Browning BLR made changes on the receiver tops in 1995/96. The older receiver tops on pre-1996 BLR rifles were flat. The new BLR's made from 1995/96 on, have a semi-round top receiver. So make sure when your BLR was made. Older pre-1996 mounts from any other manufacturer can't be used on the newer BLR's either.
The primary difference between the original BLR and the '81 BLR is in the receiver shape and, consequently, the magazine. The original BLR (1970 - 1980) has a concave-shaped receiver - it is wider at both ends (flared out), where it attached to the stocks, than it is in the middle. Consequently the magazine is narrower and thus had to be made longer in order to hold an adequate number of shells. The 81 BLR's receiver is straight-sided, which allows a wider and thus shorter magazine which is, more or less flush with the bottom of the receiver. The Lightning has a flare in at the front of the receiver. The Model 81 Lightweight has flare in at the front and rear of the receiver. The Lightweight Takedown has a flare in at the rear of the receiver.
Having owned both, I have not developed a preference over the extended or flush magazine. Some people dislike the extended magazine, but I remain neutral on the issue. The only advantage is that the flush magazines are readily available, and they are the same and interchangeable between model changes since the Model 81’s became available.