Long time ago...
Lyman took over ideal and continued the hand tool dies. 310 tool. "Nutcracker." Haven't looked to see if still made for some time. And Lyman offerred a press for this tool/dies. I think the die diameter was 3/4. I just barely remember these presses available when I was young 50 years ago. Lyman offers/ed an insert for other presses so they could use these smaller body dies... I still have some in the plastic, somewhere...
Since, 7/8 has been standardized. Unless you are looking shotgun or .50 Cal Browning... then you might see some 1 1/4 or 1 1/2 dies. That is why there is a bushing in the Rockchucker Press and some others. Lachmiller used to offer shotgun dies... Slow... but they work.
So, if you are talking standard dies in standard presses, yes, they pretty much interchange. If you wander off to play with a .50 Cal... ask carefully. It is available, but costs alot more and does not interchange. happy holidays.