For the case of Strongbow at the very least, its worth keeping in mind that, similar to a lot of the Norman campaigns in places like Italy as well, it was simply a stroke of luck that came from specific circumstances the Anglo-Normans didn't take the initiative in.

Diarmuid MacMurrough was a king of Leinster who had been defeated and expelled from his lands by Tigernán Ua Ruairc (in Breifne) who had the support of the High king Ruaidrí Ua Conchobair. As a result of this MacMurrough sought help from Henry II of England, he didn't have time for such things personally but allowed MacMurrough to recruit mercenaries and help in his domains. This led MacMurrough to the Wales and the areas that had been conquered by the Anglo-Normans relatively recently around southern Wales. This area was obviously physically close to Ireland and had a lot of economic and cultural interchange, especially near Bristol that where a lot of the trade from Ireland passed through. A lot of the Marcher lords would have had good, recent combat experience from the intermittent Welsh wars (Henry II had launched multiple invasions of Wales which achieved nothing). Richard De Clare, AKA, Strongbow was the Earl of Pembroke, and he did not have good relations with Henry II, his family had been erratic about supporting Matilda or Stephen during the Anarchy, with Henry perceiving him as disloyal against him and his mother, so he withheld on granting him the title of Earl of Pembroke despite the death of his father. Some of De Clare's holdings in France and England had been lost over the years from the war and the bad relationship with the king. Suffice to say, Richard De Clare was at something of a low point, and MacMurrough's arrival presented him a golden opportunity to win for himself a much better position. Through negotiations Strongbow was able to prompt MacMurrough to grant him his daughter in marriage and the succession to the kingdom of Leinster, in exchange for military support to win back MacMurrough's lands and titles. This was much more attractive than anything that could be expected against another, likely unsuccessful campaign against the Welsh, and after waiting for two years for Henry II to sign off on allowing him and the other Anglo-Normans to intercede in Ireland the stage was set.

In addition to De Clare there were various other petty lords that were recruited in this first stage of the conquest, promised various other spoils such as the productive town of Wexford to Maurice FitzGerald and Maurice De Prendergast. Interestingly a lot of these men had direct ties to the Welsh as well as the Anglo-Normans, including Strongbow's second in command, Raymond "Le Gros" FitzGerald. He was a grandson of Princess Nesta, daughter of Rhys Ap Tewdwr, the king of Deheubarth. She had a lot children by a lot of different men, including a son who also partook in this first invasion, Robert FitzStephen. A lot of the men that would have been brought to bear in this expedition would have been Welsh as well, especially archers who were particularly crucial in the battles that came. I think this all shows the very complex identity that worked in this whole affair, the Marcher lords were simultaneously Welsh by blood and often culture while trying to subjugate the independent Welsh kingdoms.

The first wave was led by Robert FitzStephen in 1169 who only had a small force of about 450 men but managed to take Wexford and raid into Leinster. Initially the High king Ruaidrí just let MacMurrough return to his old position of king of Leinster with an agreement at Ferns that MacMurrough recognize him as his superior and get rid of his new allies after threatening him with a show of force. He doesn't seem to have fully understood that this was more than just the usual petty warfare between local rulers in Ireland, and instead a preamble to a major invasion from Britain. MacMurrough seems to have gotten notions that, with the help of his new friends from Wales, he'd be able to go further than simply taking back his own lands but instead challenge Ruaidrí for the High Kingship. In 1170 Strongbow himself arrived (screened by Raymond Le Gros) with a much large force of more than 1200 men and they sprang into action. Waterford was besieged and taken. Dublin agreed to recognize MacMurrough's overlordship seemingly without too much fighting. Its important to note that most of the towns in Ireland at the time were established by and controlled by Norse-Irish elites who were culturally and politically distinct from the rest of the country, the lords who controlled Dublin acted independently and wouldn't have had much reason to respect the authority of Ruaidrí Ua Conchobair, but would prefer not to get into a major showdown with MacMurrough and his seemingly invincible new mercenaries. Strongbow officially wed MacMurrough daughter, Aoife, at Waterford which solidified his new claims in Ireland and gave him and the other mercenaries all the more incentive to see this whole affair through. Ruaidrí Ua Conchobair seems to have then realized the true danger of the situation and marched to properly engage MacMurrough and the Anglo-Normans. Unsurprisingly he was less than impressed with MacMurrough reneging on the Ferns agreement and executed MacMurrough's son who he had taken as a hostage to ensure cooperation. Dublin rebelled against MacMurrough in the face of Ruaidri's advance, but it was quickly put down.

Finally, before any major engagement with Ruaidrí, MacMurrough died. This changed everything, as had been agreed upon Strongbow was now the lord of Leinster, but such an agreement was not legal in the eyes of Irish law and opposed by other members of MacMurrough's family. Its at this point that it really becomes a proper invasion of Ireland and not just a mercenary venture. Through the skin of his teeth Strongbow was able to fend off uprisings all across his newly won lands, and Ruaidrí Ua Conchobair's attempted siege of Dublin. The Norman cavalry and Welsh archers have been traditionally credited with being very difficult for the mostly foot infantry Irish troops to handle.