Obviously Pool of radiance was first and probably the most primitive. Do the later games get more sophisticated in terms of the interface or does it say mostly the same?

Do they ever gain mouse support? Music? The ability to move characters in and out of the party?

The only game from the collection I'm truly familiar with is Forgotten Realms Unlimited Adventures, which is the last title, but not really part of any series, it's just a tool to create and play your own Gold Box adventures with. It's based on the most advanced version of the engine, and it has all of the features you're looking for.

So yes, there is progress in the interface. I'm not perfectly sure but it might have started around 1990 already. IIRC, Gateway of the Savage Frontier has mouse support and music and maybe even that sort of party managment you're looking for. And I see a mouse cursor in screenshots of Death Knights of Krynn, too. Not sure about Secret of the Silver Blades.

Found this list someone made and added the dates from Wikipedia:

FORGOTTEN REALMS:
Pool of Radiance (1988, EGA / PC Speaker)
Curse of the Azure Bonds (1989, EGA / Adlib)
Secret of the Silver Blades (1990, EGA / Adlib + Roland)
Pools of Darkness (1991, VGA / Sound Blaster + Roland)

DRAGON LANCE:
Champions of Krynn (1990, EGA / Adlib)
Death Knights of Krynn (1991, EGA / Adlib)
Dark Queen of Krynn (1992, VGA / Sound Blaster + Roland)

SAVAGE FRONTIER:
Gateway to the Savage Frontier (1990, VGA / Sound Blaster + Roland)
Treasures of the Savage Frontier (1992, VGA / Sound Blaster + Roland)

So it seems like Gateway to the Savage Frontier is the last game from 1990 and the first to switch to VGA graphics, so it's quite possible that this is the turning point, and that all games after it have a bit more modern interface with mouse support.

Well there is a Gold Box Companion which should ease the pain, I never use it but I plan to complete Gold Box games one day.
http://gbc.zorbus.net

FEATURES
HUD above DOSBox window with hit points, character icons, XP meters and effects. Shows good effects in green, bad in red. Also shows if a character is level drained.
Fix-command for Pool of Radiance. Works with the other games as well. Instantly heals the characters. Optionally also fixes level drained characters. Can not be used while in combat.
Temporarily change demihuman race to human to avoid experience level limits.
Optional level up without visiting training halls and without race-/game-related level limits.
Store the list of memorized spells and restore the spells with a single click. Can not be used while in combat.
Automap with party location. Notes can be added to the map.
World map view. Shows party location in some games.
Combat view. Shows character / monster locations. Shows if character/monster is held or helpless.
Journal entries.
Optional auto-identify of items.
Optional auto-ammo increases the amount of arrows / bolts / darts to a wanted level.
Autodisable quickfight after combat. Some of the older games didn't disable quickfight between fights so you could enter a dangerous fight with quickfight on.
Character editor. Reads/modifies memory so it's instant compared to save file editors. Can not be used while in combat.
Teleporting. You can teleport to a chosen location on the map. You can also teleport on the world map in some of the games.
Save Game Editor. You can edit party composition (swap members in / out), edit inventories and effects.
Some modding tools for changing the font and editing game data.
Lists saved games chronologically with a description of the location (map name).
Able to backup to latest save game with a description.
Playing paladin or ranger in Pool of Radiance. Can be exported to Curse of the Azure Bonds.
Very experimental monk class for Pool of Radiance. Can be exported to Curse of the Azure Bonds.

They add some much needed interface improvements along the way. The Fix command for automatically using your heals and memorizing them again during camp. Automatic preparation of the spells you chose last time when resting. Some redundant steps when navigating the interface get removed. Slicker input response. Mouse support. Adlib/Sound Blaster support and eventually VGA colour support.

Could you please elaborate, especially the VGA part?

VGA supports 256 colour palettes, whereas the earlier games use EGA, which only supports 16 colour palettes. Compare the graphics of Pool of Radiance with The Dark Queen of Krynn for reference. Adlib/Sound Blaster were music synhtesizer/sound cards, and these allowed for better sound effects and music.

I see, another confusion. I thought you meant that mysterious "they" added VGA support to old EGA games. No matter.