Field (RA & RHA) as opposed to siege
Of course they did for example
Alma - 1 troop of RHA attached to the Light Brigade but each division had their RA batteries
Inkerman- 7 RA batteries. In fact the batteries were rallying points and Russian objectives.
They are often missed off orders of battle.
With Sevastopol the British field artillery esp RHA was deployed to face threats from the East which never really materialised after Inkerman. The French and Sardinians deployed their field artillery at the battle of Chernaya so the threat did materialise but in a different sector.
Royal Artillery and Royal Horse Artillery guns were towed by horses the latter at a cavalry pace. I am not sure in a battle like Inkerman where the horses were after the guns had been emplaced as they had to be reasonably close but not close enough to be shot. Smaller RA field pieces could be manhandled and chains and ropes were provided for this and infantry could be detailed to help . Short distances or minor changes of position therefore could be man powered but horses had to be on hand. RHA horses were always kept close but still quite a distance from the guns. RHA often had spare teams towing a limber and no gun. All of the field artillery sets are short of horses.