It's in William Shakespeare's text, but during the English lesson between Katherine and Alice, Katherine switches from the familiar "tu" (which is to be expected when a princess addresses a servant) to the formal "vous" (which would be appropriate for a younger person addressing an older one, were the younger one not of royal blood) in her first line of dialogue in this scene, and continues using it throughout thereafter.
The French herald, Montjoy, is with Henry V when he receives the lists of the dead. This is immediately followed by the tracking shot of Henry walking over the battlefield, part-way through which he passes Montjoy who bows to him. Unless Montjoy did some pointless off-camera sprinting, he could not have got ahead of Henry in time.
Despite the exciting battle scenes the whole English army is thought to have fought on foot, so knights would not have engaged the French cavalry from horseback as depicted. Large numbers of the French army were also dismounted although cavalry attacks were also launched against Henry.