First Phase: The initial crisis, re-inforcement of the Legations, Admiral Seymour's abortive attempt at a relief expedition, would see the vessels listed in an earlier post referencing the Taku Forts bombardment website :
especially the smaller gunboats/TBDs.
Second Phase: The Taku Forts operation, Tientsin, and the successful relief of Peking under Gaselee, would see a large build-up from all nations, perhaps where vessels were arriving from further afield with troop re-inforcements.
Third Phase: The 'mopping up' operations under Count von Waldersee: This would perhaps see the relief of vessels earlier on station, and you could definitely point to a build up in the German presence, i.e, the Battleships Brandenburg and Worth.
I'm afraid I'm not really confident enough to break down when each ship came and went; this isn't a doctoral thesis afterall!
Anyway, without further ado:
Protected Cruiser: D'Entrcasteaux: own class, N2324/23a
Descartes, Pascal: Descartes Class, N2326
2nd Class Cruiser: Jean Bart: Alger Class, N2328
GunBoat: Lion or Leon (?): class unknown, no direct Navwar equivalent.
Great web page regarding the French Squadron prior to 1900:
Battleship: Oregon: own class, (Indiana/Massachusetts), N6123
Protected Cruiser: Newark: San Francisco Class, N6330/31
Nashville: own class, no Navwar equivalent
Unprotected Cruiser: Yorktown: own class, no Navwar equivalent
Despatch Vessel: Monocacy: double-ended side wheeler, launched 1866, use N4711 Esplatore
Armoured Cruiser: Vettor Pisani: Carlo Alberto Class, N4308
Protected Cruiser: Stromboli, Ettore Fieramosca: Etna Class, no Navwar equivalent
Calabria: own class, N4309/25
Elba: own class, N4324
Armoured Cruiser: Kaiser und Konigin Maria Theresa: own class, N8133
Kaiser Karl IV: own class, no known Navwar equivalent
Protected Cruiser: Kaiserin Elisabeth: Kaiser Franz Josef Class, N8138
Zenta, Aspern: Zenta Class, N8137
Battleship: Brandenburg, Worth: Brandenburg Class, N3112
Armoured Cruiser: Kaiserin Augusta: own class, N3330/31
Protected Cruiser: Hansa, Hertha: Victoria Luise Class, (as Freya), N3327
Gefion: own class, no Navwar equivalent
Irene: own class, no Navwar equivalent
SeeAdler: Bussard Class, no Navwar equivalent
GunBoat: Iltis, class unknown, no Navwar equivalent
Battleship: Petropavlosk: Poltava Class, N7110
Cissoi Veliki: own class, N7111
Navarin: own class, N7112
Armoured Cruiser: Rossiya: own class, N7309/10
Dimitri Donskoi: own class, N7322a
Vladimir Monomakh: own class, N7323a
Rurik: own class, N7311/20
Protected Cruiser: Amiral Korniloff: own class, N7319/24
Sloop: Razboinik, Vjestnik: Kreisser Class, no Navwar equivalent
GunBoat: Sivutch, Bobr: Sivutch Class, no Navwar equivalent
Korietz, Mandjur: Mandjur Class, I used the Khabry, N7701
Armoured Gun Vessel: Gremyaschi, Otvajny: Grozyaschi Class, no Navwar equivalent
Torpedo Gunboat: Vsadnik, Gaidamk: Kazarski Class, I used Abrek, N7702
Gilyak: own class, could probably use the Abrek as well, or perhaps the Puilki, N7509.
Phew, that completes things so far, although of course a cogitation on the Chinese vessels that may have been around is forthcoming, as well as some more photos of the Russian and Japanese contingents done so far, so watch this space!