The situation seems comparable to the Roman and the barbarians; would the native inhabitants have agreed? Was there a difference between Islamic and Christian perspectives, or were they regarded as outside invaders by both groups?
I've answered quite a similar question here: https://www.reddit.com/r/AskHistorians/comments/9cz69o/when_the_crusades_were_called_by_the_pope_what/
The short answer is that most of the native population did not seem to mind. William of Tyre claimed that most of the Franks integrated well by learning Arabic. The 1120 Council of Nablus identified that intermarriage was common and that many European settlers wore Islamic dress. The council tried to ban these things since the leadership were concerned that the Franks were losing their Christian identity (a common criticism of the Crusader States back in Europe), but these things were difficult to enforce and they continued anyway. The native people there did not initially think of the Franks as an existential problem. That is not to say they liked them, the massacre at Jerusalem in 1099 caused outrage, and many Islamic knights wrote many poems about it. But the idea that the crusaders were an existential threat was a fringe view. Those who did view the crusaders as a problem were quickly placated by lower taxes (the Fatamids had been collecting 70% of produce, whereas the crusaders wanted 50%) and trade oppurtunities - there is no mention of any civil unrest under early Christian rule in either Christian or Islamic sources.
Some even liked Christian rule. When the crusaders arrived, the rulers of that area were the Fatamid Caliphate. The population of the Holy Land was mostly Sunni, whereas the Fatamids were Shi'ite. Many saw the crusaders as liberators rather than conquerors and on the march from Antioch to Jerusalem the crusading army found that most towns happily opened their gates and set up markets.
On the political level, although the surrounding states were unnerved by the arrival of this large army, most were too busy fighting each other to mount any kind of defence and it frankly it wasn't worth it for them. The interest of the crusading army was plainly the western coast, which held little strategic value to most Islamic factions. At the time, Jerusalem was not an especially important city since Madina and Mecca were far more spiritually valuable. Many states quickly realised the trading oppurtunities that the First Crusade opened up and were keen to trade, particularly Damascus. The exception to this was if the crusaders pushed east. Incursions east of the Dead Sea were met with resistance, and an attempt to expand toward Aleppo was crushed at the Battle of the Field of Blood in 1119, but for the most part the Crusader States were just another few small principalities among many others.
In the 1130s and 40s, however, there was a new wave of Frankish immigrants which did not integrate. As they saw it, they were entering a Latin, Christian kingdom and didn't get on with the locals. Both William of Tyre, a high ranking bishop in Jerusalem, and Usama Ibn Munqidh - a diplomat and nobleman affiliated with Damascus - attest that these new arrivals stirred up trouble. Usama records an instance where one of his trade negotiators was threatened by a mob of Franks, only to be chased off by one of the original crusading knights. By the time of the Third Crusade, the region had been thoroughly polarized, which colours historical accounts. The Islamic world had also stopped its infighting and had been unified by Saladin. Saladin's secretary, Ibn Al Athir, wrote that the Third Crusade was viewed as an existential threat. Richard the Lionheart terrified the Islamic nobility, since he was the only person who ever defeated Saladin in battle. Not only did Richard beat them, he humiliated them. At the Battle of Jaffa, Richard used 2000 crossbowmen to kill almost all of Saladin's cavalry before they even reached Richard's men. Then Richard got on a horse and gallopped down Saladin's battle line taunting them - not one of Saladin's men responded to Richard's taunts and Saladin retreated, deeply ashamed and angry at himself for losing so badly. Richard was to the Islamic world as Hannibal was to the Romans. However, in the 13th century, European military expeditions had been outclassed by the Mongols as the ultimate threat to Islam.
The ones who viewed the crusaders as barbarian invaders were the Byzantines. Anna Comnena, a Byzantine princess writing about the First Crusade, used the word barbarian frequently and sometimes referred to them as Kelts and Gauls as well. Most Greek accounts of the crusades considered them to be barbarian invaders. The pro-Byzantine historian Stephen Runciman (not a medieval author, he was writing in the 1950s and 60s) described the crusades as 'the last of the barbarian invasions'.