There doesn’t appear to be a James Murphy of a cavalry regiment in the Crimea casualty lists; what is the key data you have - regiment and/or war or date of death?
To try and add some detail to the name, I’ve checked the actual medal roll (hard copy) for the Crimea Medal & Clasps awarded to the 6th (Inniskilling) Dragoons, but failed to pick out any James Murphy, I also checked to rolls for 5th Royal (Inniskilling) Dragoon Guards & the 6th Dragoon Guards (Carabiniers) with nil return.
You should check the database of the UK, Military Campaign Medal and Award Rolls, 1793-1949 that is on Ancestry, for his name in case I missed it and to confirm his regiment.
Also be aware that 30 May 1854 the 6th (Inniskilling) Dragoons embarked on HM transport Europa at Devenport departing for Bulgaria, enroute the vessel “caught fire and there was a great loss of life” and one would also expect that many others would have been injured.
There doesn’t appear to be any surving casualty returns (lists of non-effectives; sick, wounded, deserted, discharged & died) for the 6th (Inniskilling) Dragoons in the National Archives Kew for the 1853-5 period.