In theory, Magna Carta had guaranteed that peers would always be tried by their peers. However, there were exceptions to this, certainly in the fifteenth century. One day when I have more leisure I will discuss the relevant precedents - there are quite a number and it would make for a long and (possibly) tedious post. Anyway, the theory was that peers would be tried by their peers - let us say for convenience this means before the House of Lords, making use of Common Law procedures. I should add though that the peer being tried was not allowed to have a counsel to present his case, and nor was he given advance notice of the nature of the prosecution case or who the witnesses (if any) would be. So we are not talking about what the 21st Century would call a fair trial. The peer would have to answer the charges on his own wit, and, to be frank, if matters had got this far his chances of acquittal were slim. Off hand I can only think of one fifteenth century peer who was acquitted - Northumberland in 1403, when he was pronounced guilty only of trespass. The circumstances were particular. Henry IV's political position was weak, and he had many enemies among the peers. He made sure thereafter that no peer received a trial before Parliament during the rest of his reign.
Although Richard II had empowered his Constable and Marshal to 'arrest and chastise all traitors' the actual use of the Constable's court for this purpose was rare up until the Yorkist period. The most common use of the court was in the immediate aftermath of a battle, where it was felt that summary justice was appropriate. The Constable's Court did not use Common Law, but rather the Roman or Civil Law. Under this system it was the duty of the judge to examine the 'facts' and decide on the guilt or innocence of the party without reference to a jury. Of course, the Constable (and/or Marshal) was normally, if not invariably, a peer, so in that sense the accused still received 'trial by his peers', but not, I think, in the sense originally intended. Roman or Civil Law took much less cognisance of individual rights and was more about enforcing the power of the sovereign. This was never more true than in treason cases.
Let us be quite frank. The Constable's Court was a kangaroo court. You had about as much chance of being acquitted as you had of flying to the moon. Just because it was called 'a court' it does not mean it was anything that people living in a modern democracy would recognise as such. It was a formalised lynching party.
Now, in the immediate aftermath of a battle, where no one could sensibly deny that Lord X had made war upon the King, this was maybe no big deal, away from those more concerned with theoretical rights than practicalities. But to use it in other circumstances was harsh and novel.
To discuss the fate of the Earl of Oxford, we must return briefly to 1462 when, as you may remember, Margaret of Anjou was in France planning an invasion of England. She wrote to Oxford and Oxford wrote back. Unfortunately for them both (but particularly for Oxford) Yorkist Intelligence intercepted the letters, opened them for inspection and copying, and then allowed them to proceed. Yes folks, there really was a Yorkist Intelligence Service, though it was not officially known as such! Espionage and the monitoring of individuals for possible subversive action was another thing not invented by the Tudors.
The Government waited until details of Margaret's landing were revealed by the correspondence, and then John Tiptoft, Earl of Worcester - the Constable - aided by Lords Ferrers and Herbert arrested the alleged conspirators, namely Oxford, his eldest son, Aubrey de Vere, Sir Thomas Tuddenham, Sir John Montgomery and Sir William Tyrrell. All were promptly tried by Worcester in his 'Constable's Court' - Ross says 'speedily convicted'. Arrested on 12th February, the last of them died on Tower Hill on 26th February.
Now, there had not been a battle and nor was there any immediate danger to justify this haste. Oxford at least ought to have been tried by the House of Lords, and there is really no cogent reason why he could not have been held in the Tower until they were assembled. The 'process' - if it can be so dignified - was unprecedented in a case of conspiracy, and it's a neat question whether Oxford's dealings actually constituted treason. (Under Henry VIII they certainly would have done, but the law of treason in 1462 was much more tightly drawn.)
In effect, Oxford was 'murdered'. But I have never come across anyone calling Edward IV a murderer on his behalf.
Oxford's younger son, John de Vere, was eventually allowed to inherit the title, and he married one of Warwick's numerous sisters. The treatment of his father and brother turned him into a lifelong 'Lancastrian' and despite various escapades he was able to survive to fight for Henry Tudor at Bosworth.