On PlayStation, saves do not live in the game but on a memory card — a file, as far as emulation is concerned. This page explains where those files are, how to add a second card, and how to import saves coming from elsewhere.
With the Libretro cores (PCSX-ReARMed, Beetle PSX, SwanStation), memory cards and save states live in:
/recalbox/share/saves/psx/
The port 1 memory card is named after the game, with the .srm extension. With DuckStation as a standalone emulator, cards live in a dedicated subfolder:
/recalbox/share/saves/psx/duckstation/memcards/
Back this folder up regularly: it is the only place your progress lives.
Some games need a second card to copy or import data. Unlike the first one, it is shared by every game.
HOTKEY + B, then enter Options (the core options).Enable second memory card option.
The second card then appears in the game's slot 2, and a pcsx-card2.mcd file is created in /recalbox/share/saves/psx/. This has to be repeated for every game concerned: core options are set per game, unless you save a global option file from RetroArch.
The console BIOS contains a manager that copies and deletes save blocks. To reach it:
HOTKEY + B, then Options.Show Bios Bootlogo option.HOTKEY + left stick up.You land in the PlayStation BIOS interface, where you can manage your save blocks — and, if you enabled the second memory card, copy data from one card to the other.

.mcr files)ePSXe memory cards are already in the right format, only the name changes:
ePSXe/memcards/..srm extension — for instance Final Fantasy VII (Europe) (Disc 1).srm./recalbox/share/saves/psx/..gme files)DexDrive saves must be converted first:
File > Open, then select your .gme file.File > Save as…, and save in the .mcr format, using the exact name of your rom..mcr extension to .srm and copy the file to /recalbox/share/saves/psx/.The save file name must match the rom name exactly. One capital letter or one space off, and the emulator will silently create a blank card next to it.