Well, you seem to be referring to medieval warfare but I can comment on the Napoleonic Era.

Yes, parties did storm the walls with ladders. In many cases cannons fired at a section of wall to create a breach which was little more than a mound of rubble leading up to the top of the wall. This could take a very long time and consume huge amounts of ammunition before the breach was deemed practicable. This was normally very difficult to climb and if it was too steep would mean the soldier had to put his musket on his back to climb with both hands. Leaving him defenceless and unable to return fire to the defenders on the wall. The results of such direct assaults were incredibly brutal and consequently were fairly rare.

Assault by escalade (using ladders) was also common, although often used in combination with a breach. They were often used as diversionary attacks whilst the main breach was assaulted to force the garrison to divert its manpower. Ironically enough at one of the most bloody direct assaults of the peninsular war, the siege of Badajoz, the assault on the breach failed but the diversionary attacks succeeded.

The initial assault was called the forlorn hope and (for the British army at least) was composed mostly of volunteers. It was probably the most dangerous operation it was possible to perform in the military of the day. The name comes from the Dutch Verloren hoop meaning 'lost party', the French equivalent was les enfants perdus meaning ' the lost children'. It was basically choosing to die first.

However there was no shortage of volunteers, which is odd, the rewards of being in the forlorn hope were large if you were a senior NCO or an officer, as it resulted in promotion and for the NCO's a commission. To such an extent that there was fierce competition between officers for the honour of leading the forlorn hope. For the rank and file there was no such reward yet they never lacked volunteers and in the case of the 51st at Badajoz there were so many volunteers that the stormers were chosen by the regimental adjutant at random. (He walked down the line blindfolded and touched men at random.) Why there were so many volunteers is mind boggling but even ignorance was not an excuse as many who survived their first forlorn hope volunteer again the next time around.

In summary, even in an age of gunpowder where cannister shot could shred a dozen men in a second and the survivors of a forlorn hope would describe the breach as 'the mouth of hell', where mines of barrels of gunpowder and nails through pieces of would were scattered in the breach, assaults up a breach in the wall or assaults by ladder were still a part of siege warfare.

The Napoleonic Source Book -Philip J Haythornthwaite

Die Hard! Dramatic Actions from the Napoleonic Wars -Philip J Haythornthwaite

Pardon me, but what do you mean by 'assimilated'?

For the Hundred Years War, fortifications were often taken this way- but not openly in broad daylight. By stealth, at night, when it was hard to see ( this is before searchlights). Smaller castles, fortified houses, etc. could often not maintain a constant watch over every place on the walls. A band of routiers could make a surprise assault on an isolated castle and often take it.

Sumption: The Hundred Years' War