Remove slide. remove the barrel and recoil spring/guide.
1. Find a strong buddy to push down on the front of the sight rail with a chunk of dowel rod or something else that is a little more narrow than the two mounting shoulders. , (above the cross pin) until it lets you drive out the pin. Right to left I think... See if one end is larger and drive from the smaller end. Then push down on sight rail while pulling it backwards.
OR... 2. Use a plastic jawed C clap or cabinet clamp and a small piece of wood over the front of the sight rail to depress it enough to drive out the pin IF there is a pin in the Inglis versions. The problem is that the C-clamp can get in the way. I think that I ended up padding my small vice and used part of a toothbrush handle to depress the sight as it sat sideways in the vice jaws.
I heard that they may have made some with a fixed pin ( well really studs) that only needed to be pressed downwards and then pulled to the rear once they were free of their channel. I do not know if this is really true...
I have a Chinese contract Inglis (2CH) with a tangent rear sight. I replaced it because the rear sight adjustment slider had been damaged by a prior owner, who then used JB Weld to fix it in place (I know, ugh.) I used Float Pilot's February 7, 2017 post and scythefwd's February 18, 2017 post to figure it out. Float Pilot's instruction No. 2 seems to be correct, as I can find no other information about there being a pin to remove. I was able to find the leaf at one place (Numrich/Sarco/Midway) and the rear sight adjustment slide somewhere else (same). The parts were $87 (Oct. 2018), although in retrospect I really could have re-used the leaf and only needed the adjustment slide, which was $12.
So, to add to Float Pilot's and scythefwd posts:
The tangent rear sight leaf pivots up and down at its front end on two studs, one on each side of the leaf.
There is no pin to drive out. Under the leaf is a flat leaf spring about 1/4 inch wide that fits in a lengthwise slot on the underside of the leaf. (See Float Pilot's photo above.) It pushes the front end of the leaf up flush with the top surface of the sight mount. (See photo below.)
Inside the front end of the rear sight mount, on either side, are horizontal lengthwise grooves that the two studs on the leaf can trace to get the leaf off. But to do so, the front end of the leaf must be depressed as far as it can go (maybe about 1/8th inch) to line up the studs with the grooves.
So, in order to replace the leaf, you must depress the front of the sight leaf as far as it can go until the studs line up with those grooves, and then pull rearward on the leaf to get it off. I used a 1-inch long piece of 1/4 square dowel (that I bought at Home Depot) and a 1-inch cushioned spring clamp to push down the leaf at the front end until the studs lined up with the grooves, and then pulled back on the leaf and it came off. (See photo below.) You may have to add thumb pressure to get the leaf all the way down. I had to carefully clamp the slide in a vise (be careful you do not crush the slide) and then use the dowel/spring clamp rig to push down the front end of the leaf to position the studs with those grooves, so the leaf could be pulled backwards. It took a couple of tries to get the leaf to budge (it's been there for over 70 years) but eventually it cooperated. You may need pliers to pull back on the leaf. I think it's might be possible to do this without a vise but it probably would be awkward. Be careful when pulling back on the leaf that the spring clamp does not pop off and hit you in the face (boiiiing).
Installing the new leaf and slider involved transferring the innards of the slider (a smaller diameter piece and a tiny cone-shaped spring) to the new outer piece, sliding the sider into position, transferring the leaf spring from the old leaf to the new, and then sliding the leaf back into place by lining up the studs with the grooves and pushing the leaf forward. It snapped right into place.
I suppose these instructions would also apply to a Browning Hi-Power with tangent sights, which appear to be identical.