Before digging into your exact question, let's get cover some background on Moray and Wallace.
Andrew de Moray, also known as Andrew Murray, was born to Andrew Moray of Petty circa 1270CE, although the exact date is unknown. The original Moray/Murray progenitor is believed to be a man of Flemish origin who settled in Scotland in the 12th century [2] as part of David I of Scotland's efforts to make the Scottish church and government more in line with that of mainland Europe. The de Morays in particular were brought in to help David (and later his successors) pacify the region, which had resisted joining the newly consolidated Scottish state [1,3]. By the 1340s, under Alexander III, the Murrays had accumulated both significant political power, a large network of land holdings, and considerable personal wealth (to the extent that Andrew Murray's uncle William acquired the nickname le riche among the Norman nobles) [3]. Due to this history, the Murrays had strong financial and political motivations to resist Edward I's rule when the English monarch invaded in 1296.
In contrast, we don't know a lot about William Wallace's background. A letter with Wallace's personal seal, preserved in the Lübeck city archives, indicates that his father's name was Alan Wallace. He had at least two brothers, John and Malcom, and his father was a vassal of James Stewart, High Steward of Scotland. Based on his sword, and contemporary descriptions, he was a large man who stood about 6'7".
In the spring of 1296, Edward I invaded Scotland to punish the Scottish king, Edward Balloil, for refusing to acknowledge Edward as his superior. After the English forces sacked the port town of Berwick, Balloil led the Scottish forces to meet the English in battle at Dunbar in April. The Scots were soundly defeated. Andrew Murray, fought at Dunbar alongside his father; both Murrays were captured and imprisoned [5, 3]. The senior Murray died in the Tower of London. Andrew Murray was held at Chester Castle, but escaped in early 1297 and returned to Scotland. Upon arriving at his old family lands in Moray, Murray publicly proclaimed his defiance of English rule. The Scottish public had already turned against Edward I due to the killing of civilians at Berwick and the high taxes imposed on the Scottish population, so Murray was able to quickly raise a guerrilla army and start a revolt [6]. By July 1297, William FitzWarin, an English official based at Urquhart Castle, was alarmed enough by the uprising to write to Edward I that "[s]ome evil disposed people have joined Andrew Moray" and were in open rebellion [3].
Meanwhile in Lanark, William Wallace started a riot in a court being held by William Heselrig, Sheriff of Lanark (one of Edward I's appointees). It is not clear what motivated Wallace to start the fray. There is some speculation that it was in coordination or solidarity with other Scottish rebels; other accounts posit that Wallace had been outlawed for murder of English officials and was captured whilst in Lanark, but there is no documentation as to what sparked the violence at the court. Other Thomas Gray, transcribing his father's firsthand account of the incident, describes Wallace as escaping from the riot with the help of his wife, Marion Braidfurte, who was then captured and executed by the Sheriff [6]. Whether or not Braidfurte was involved, Wallace assembled a band of men who killed Heselrig in his bed and set several fires before fleeing.
Due to the unpopularity of Edward I's rule, small uprisings were breaking out across Scotland. Although Edward I responded by sending troops and extracting pledges of loyalty from a number of Scottish nobles in July 1297, he was rapidly losing control of the country. Wallace set up a base of operations in Selkirk, which allowed him to accumulate more supporters. By the late summer/early fall of 1297, Murray and his forces had moved south to join Wallace. While we do not know exactly when they met in person, Murray and Wallace appear to have coordinated their pursuit of the English army, eventually reconvening when they besieged Dundee Castle together in early September 1297 [8]. When news reached Dundee that English troops commanded by the Earl of Surrey were on their way to Scotland, Murray and Wallace jointly left Dundee in the hands of Alexander Scrymgeour and advanced to Sterling as a combined force.
The letter containing Wallace's seal, which was sent in October of 1297 (after the Battle of Sterling Bridge), identifies "Andrew de Moray and William Wallace" as "leaders of the kingdom of Scotland and the community of the realm" and signed by both, which indicates a high degree of cooperation. Unfortunately, we have little in the way of personal correspondence from either Murray or Wallace, and lack contemporary accounts from eyewitnesses about their personal interactions.
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