This answer relates just to Irish and Scottish crusading.

Though the English made up the majority of crusaders from the British Isles, the Irish and Scots also participated in the crusades.

The first record of Irish crusaders is from the early twelfth century, sometime before the Second Crusade (1147-9), when a group of Ulster nobles stopped at the abbey of Regensburg en route to the East. The Anglo-Irish, that is, the English settlers in Ireland arriving from 1170 onwards, also crusaded. Anglo-Irish nobles joined the Third Crusade, such as Bertram de Verdun lord of Dundalk and Gilbert Pipard lord of Ardee. There was also Anglo-Irish participation in the Ninth Crusade, such as James de Audley, justiciar of Ireland.

The Anglo-Irish also contributed to the crusade by donating to the military orders. The Templars, Hospitallers, and Knights of St Thomas were all given lands in Ireland, the revenues of which funded the orders' campaigns in the East. However, most of the Gaelic Irish rejected the military orders, as they often fought the Gaelic Irish on behalf of the English. The Hospitallers, for example, were given Kilmainham outside Dublin as it was a defensive position to guard the city against Irish attacks from the west. Hardly any Gaelic Irish grants to the military orders survive and almost all their houses in Ireland were founded by Anglo-Irish donors.

On the whole, crusading by the Gaelic Irish was rare, particularly after the Norman Invasion of Ireland in 1170. After this, it was mostly the Anglo-Irish that joined the crusade. This was probably because the Gaelic Irish nobility and knightly class (the natural leaders and participants of crusade expeditions) were busy fighting the English settlement in Ireland. The same drop in crusading can be seen in Scotland during the Wars of Independence and in England and France during the Hundred Years War. In both cases, when the respective conflicts ceased, crusading abroad increased.

Scottish crusaders are reported as early the First Crusade in many chroniclers' lists of the nations that joined the crusade, however, no individuals are named by these chroniclers. Lagmann, Lord of the Isles, is the only known Scottish participant of the First Crusade. Scots also joined the Second Crusade, but no individual participants are known. The Third Crusade saw participation from Scoto-Norman nobles like Robert de Quincy and probably from Alan the Steward as well (an ancestor of the Stewart kings of Scotland).

Scottish crusading became more frequent in the thirteenth century. David Rufus of Forfar joined the Fourth Crusade. The de Quincy family, the Stewarts, Bruces, and the Balliols also all joined crusades in the 1200s. At least two Gaelic Scots also went on crusade in this period, the bards Muiredhach Albanach O Dalaigh and Gille Brigde Albanach.

With the Wars of Independence beginning in 1296, Scottish crusading slowed for a time, not reviving fully until 1329. On his deathbed Robert asked that his heart be taken to Jerusalem, a task given to James Douglas. Douglas went East with a retinue of about 25 knights and esquires, but stopped off in Spain to help the king of Castille against the Moors. He was killed in a skirmish there in 1330 and Bruce's heart was returned home.

With the loss of the Holy Land in 1291, Scottish knights began to crusade elsewhere. The Teutonic Knights' campaigns in the Baltic were a popular venue. At least six Scottish expeditions went there between 1356 and 1410. Scots also joined in the Alexandrian Crusade in 1365 but don't appear to have participated in other eastern campaigns like the Crusade of Nicopolis (1396).

Baltic crusading largely ended in 1410 when the Teutonic Knights were defeated by a Polish-Lithuanian army at the Battle of Tannenberg. After this there were very few Scottish crusaders. Some joined campaigns against the Turks in Greece but only small numbers.

Like Ireland, Scotland was home to outposts of the military orders. Unlike Ireland, these orders were actually patronised by the native laity and nobility. The Templars' houses of Balantrodoch (now Temple) and Maryculter, were both founded by Scots, as were the Hospitallers' preceptory of Torphichen, the Order of St Lazarus' estate in Edinburgh, and probably the St Thomas Knights' hospital in Ayrshire.

Sources:

Eric St. J. Brooks, ‘Irish Possessions of St. Thomas of Acre’, Proceedings of the Royal Irish Academy, Vol. 58 (1956/7).

Martin Browne OSB & Colmán Ó Clabaigh OSB (eds), Soldiers of Christ: The Knights Hospitaller and the Knights Templar in medieval Ireland (Dublin, 2015).

Ian B. Cowan, P. H. R. Mackay, and Alan Macquarrie (eds), The Knights of St John of Jerusalem in Scotland (Edinburgh, 1983).

Kathryn Hurlock, Britain, Ireland and the Crusades, c.1000-1300 (Basingstoke, 2012).

Alan Macquarrie, Scotland and the Crusades (Edinburgh, 1985).