Obviously the soldiers would go on the beach, but what about those people controlling the craft? Did they stay inside? Did they go back to the ships or stay put?
This answer is taken from a previous discussion on this topic here.
For most landing craft, the trip to the beach was not a one-way journey. Landing craft crews were ordered to keep their craft afloat, and able to 'retract' from the beach, once the troops had disembarked. Typical practice for a landing was to release a kedge anchor as the craft approached the beach - for the British Landing Craft Assault, this was released 120 feet from the beach. Once the troops had disembarked, the craft could be pulled off the beach by hauling in on the anchor cable. It would then go about and return to the ship that carried it, likely to pick up another load of troops. If the kedge was not used, due to lack of space or a desire for speed, members of the crew would have to disembark and push the boat off after the troops had left. If the boat fully grounded on a sandbank, then the entire crew would have to jump off the craft and push it off the obstruction.
Where craft were damaged by obstacles, or fully grounded and could not be recovered, standard practice for the crew was to stay with the craft. They were neither trained, nor armed, for fighting ashore. However, there were exceptions to this rule. Many British landing craft were manned by members of the Royal Marines. They appear to have had a tendency to join in the land battle. A beachmaster from Juno Beach, during the Normandy Landings, complained that Royal Marine crews in his sector were neglecting their craft and heading inshore. Elsewhere during D-Day, all of the craft from No. 544 LCA flotilla were lost. Their commander, a Captain Miers, formed the crews into a scratch platoon. However, they found no way to join the fighting, and wound up returning to their parent ship. Commando operations required flexibility from those participating in them, and this included landing craft crews. During an action in support of Yugoslav partisans, Lieutenant Peter Davis won the Distinguished Service Cross for forming a search party from his landing craft crews, and setting out to locate and support a party of Commandos that had been ambushed.