This a complex issue, and doesn't have one particular answer, as 'Muslim forces' groups a number of diverse powers and regimes under one title, with wildly varying forces available to them, although for simplicity's sake I'll group them into three main factions.

The first of these factions, encountered by the First Crusade as they crossed Asia Minor, were the Turks, notably Kilij Arslan's Sultanate of Rum, although the Seljuk empire was to remain a permanent antagonist to the nascent crusader states. The Turks, as a nomadic people who had migrated west through Iran and Iraq in the previous century, often fought utilising clouds of horse archers, assisted sometimes by a core of heavier cavalry, and in the case of the more-settled Seljuks, levied infantry. Their tactics mirrored those used by the Parthians against the Romans a millenia earlier, harassing and ambushing the crusader host on the march, showering them with arrows, and then charging vulnerable or fragmented formations. However, this was a very hit and miss tactic, as well-led crusader formations could consolidate and simply push through the hail of arrows, due to their armour and shields, and if the horse archers strayed too close or we caught out of position, they could then be counter-charged by the Frankish knights. The Rumite Turks suffered heavily during the First Crusade, and were obliged to turn to scorched earth tactics due to the loss of their capital, Nicaea, early in the campaign (later retained by the Byzantines), and inability to rout a portion of the crusader host at Dorylaeum.

The second faction of note was the Fatimid Caliphate, whose power base resided mostly in Egypt, but extended up the Levant coast, including Jerusalem. The Fatimids were highly distinct from the Seljuks in their methods of making war, and employed a highly diverse army structured around various ethnic regiments, predominantly relatively lightly armed infantry, but also consisting of Turkish cavalry, and even Armenian Christians, as well as a core of Ghulams, heavy cavalry composed of slave-soldiers. As such, it could be assumed that the Fatimid army was well-balanced, but it was notoriously fragile, as while preserving a balance of different ethnic troops prevented power plays within the Caliphate, it made the field army brittle. This was proven during the Battle of Ascalon in August 1099, as the forces of the First Crusade, despite being outnumbered roughly 2-1, managed to completely rout the army of the Vizier al-Afdal in the course of a morning, killing possibly up to half of them. Nevertheless, I'm getting off track; the means by which the Fatimid army would defeat their enemy were largely similar to Christian armies of the time (excepting horse archers): mass action by infantry, with a core of heavy cavalry to deliver crushing charges, and using the large numerical superiority often enjoyed by the Fatimid army to compensate for somewhat dubious quality, and the lighter equipment loads you identified in your question.

The third faction of note is that of the Ayuubid regime. Founded by Saladin in 1171, the Ayuubids replaced the decaying Fatimid regime in Egypt, and was possessed of a dramatically different military structure, centred around the Mamluk cavalryman, in a system not completely unlike Christian-style fiefs. The Mamluks were the successor to the Ghulams utilised by the Fatimid dynasty, but on a wider scale, and formed the core of the Ayuubid army, giving it a far more professional note than the Fatimids. Mamluks were heavy cavalrymen, equipped in a similar manner to a western knight; mail shirt, helmet, shields, lances, etc., although furusiyya (equestrian) manuals extant to the era showed an even wider variety of weapons. While these formed the core of Saladin's force, Saladin also employed Kurdish and Arab infantry, and Kurdish and Turkish cavalry to supplement the Mamluks. Insofar as Saladin's wars with the crusaders were prosecuted, Saladin's force largely sought victory by manoeuvre and siege; if Saladin offered battle, it was largely from a position of supreme strength or enemy weakness. During the early years of his reign, Saladin largely avoided large scale conflicts with the crusader states while he consolidated his power base, and only once that was complete did he offer battle, seeking to lure the Jerusalemite army out into the field. Saladin's force often used positioning as a force multiplier; at Hattin, Saladin sat astride the only route to an accessible water source, prompting the parched crusaders to break ranks and fragment. Similarly, he often aimed to draw out the crusader cavalry, prompting a charge from them to 'spend' the potentially devastating Frankish charge and isolate them from their infantry. Tactics aside, Saladin's army was largely orthodox for a settled, 'feudal' (and I use that word carefully here, note the inverted commas) power - a core of heavily armoured cavalry, supported by infantry, and some auxiliaries, forming a hammer and anvil in battle that was no less heavily equipped or potent than their Frankish counterparts.

With each of the main Islamic powers in the region covered to some, albeit brief, extent, I hope this has answered your question regarding the war fighting capabilities of Muslim forces within Outremer opposing the crusaders.

Suggested Primary Material

Baha al-Din ibn Shaddad, The Rare and Excellent History of Saladin, trans. D.S. Richards (Aldershot: Ashgate, 2001)

Ibn al-Athir, The Chronicle of Ibn al-Athir for the Crusading Period, from al-Kamil fi’l-ta’rikh. Part 2, The Years 541-589/1146-1193: The Age of Nur al-Din and Saladin, trans. D.S. Richards (Aldershot: Ashgate, 2007)

William of Tyre, A History of Deeds Done Beyond the Sea, 2 vols, trans. Emily Atwater Babcock and A.C. Krey (New York: Columbia University Press, 1943)

‘The Conquest of the Holy Land by Saladin’, in De Expugnatione Terra Sanctae per Saladinum, in Ralph of Coggeshall, Chronicon Anglicanum, ed. by J. Stevenson (Rolls Series, London 1875)

Albert of Aachen, History of the Journey to Jerusalem, Volume 1: Books 1-6. The First Crusade, 1095-1099, trans. by Susan Edgington (London: Routledge, 2016)

Comnena, Anna, The Alexiad of Anna Comnena, trans. by E.R.A Sewter (London: Penguin, 1969)

Gesta Francorum et Aliorum Hierosolimitanorum, ed. by Rosalind Hill (London: Thomas Nelson and Sons Ltd., 1962)

Ralph of Caen, The Gesta Tancredi of Ralph of Caen, trans. by Bernard S. Bachrach and David S. Bachrach, (Aldershot: Ashgate, 2005)

Secondary Material

Beshir, B. J., ‘Fatimid Military Organization’, Der Islam 55 (1978), 37-56

Bowlus, Charles R., ‘Tactical and Strategic Weaknesses of Horse Archers on the Eve of the First Crusade’, in Autour de la Première Croisade: Actes du Colloque de la Society for the Study of the Crusades and the Latin East (Clermont-Ferrand, 22-25 juin 1995), ed. Michel Balard (Paris: Publications de la Sorbonne, 1996), pp. 159-66

Edbury, Peter, ‘Warfare in the Latin East’, in Medieval Warfare: A History, ed. by Maurice Keen (Oxford: Oxford University Press, 1999), pp. 89-112

France, John, Victory in the East: A Military History of the First Crusade, (Cambridge: University of Cambridge Press, 1994)

France, John, ‘Warfare in the Mediterranean Region in the Age of the Crusades, 1095-1291: A Clash of Cultures’, in The Crusades and the Near East, ed. Conor Kostick (Abingdon, 2011), pp. 9-26

Hillenbrand, Carole, The Crusades: Islamic Perspectives (Edinburgh: Edinburgh University Press, 1999)

Lev, Yaacov, ‘Infantry in Muslim Armies during the Crusades’, in Logistics of Warfare in the Age of the Crusades, ed. John H. Pryor (Aldershot; Ashgate, 2006), pp. 185-208

Lev, Yaacov, ‘Regime, Army and Society in Medieval Egypt, 9th-12th Centuries’, in War and Society in the Eastern Mediterranean, 7th-15th Centuries, ed. Yaacov Lev (Leiden: Brill, 1997), pp. 115-52.

Smail, R.C., Crusading Warfare, 1097-1193 (Cambridge: Cambridge University Press, 1956)

Why would they probably have lighter equipment?