Yes, if you were a good fighter you could get away with quite a lot. If you were careful about committing crimes fighting for the right person, you could in fact elevate your station quite considerably. Such is life in a zero-sum game warrior society.

But knights beating up on malnourished peasants was not likely to be a common judicial affair. They are not of the same social station, so the circumstances would have to be pretty exceptional for authorities to go to the trouble of organizing such a trial. These were not simple dust-ups, two men swinging at each other in a barn somewhere. The highest authorities were involved, a special ground would be prepared, there was gallery seating for the judges, and often a gallows was constructed, to execute the vanquished in case they were not slain in the combat. If champions were being used, the actual principals in the dispute were strung up on the gallows out of sight of the combat, so that one or the other could be immediately hung when the combat was decided. That's a lot of trouble to go through for a malnourished peasant. Generally one could count on being tried by one's peers, so such affairs were reserved for the aristocracy.

One of the last trials by combat in England was requested during the reign of Elizabeth I, over a land dispute in Kent. At that late date the Court found the notion a little ridiculous, but found that they could not deny it in law since the petitioners had accepted the challenge. So the court had to go through all the ritual of appointing champions, building the lists and spectator galleries, and running all the ceremonies. It was a big deal and drew huge crowds (something like 4000 people turned out), but the whole thing was a bust because the Queen intervened to avoid bloodshed, the combat never happened, and everyone went home disappointed. Which just makes the expense of the affair all the more notable.

But how would a peasant fight a knight? He has no armour or weapons to meet the knight on the knight's terms. And more importantly, he has lacks the social standing and reputation to argue his case well enough for all the higher authorities to agree that he makes a compelling point, he should be considered the legal equal of a knight, and although the truth of the matter might be quite cloudy, perhaps we could find the funds to go through the expense of a grand public trial by combat. Not likely.

As for how a knight would fight a peasant - he would probably just skip the legal niceties, and just do it. For such a matter to blow up into a legal affair that seriously troubled the knight, someone with sufficient standing would have to take the matter "over the knight's head." But who was in a position to do that? The owner of the lands that the peasant worked could do it, but in all likelihood that was the knight himself! Unless the knight is wandering off and randomly murdering people on other estates, in which case the legal dispute would end up being between some other landlord and the knight, and the peasants involved would end up being minor footnotes to the affair.

Which is not to say that peasantry were not entitled to their own comparable but more modest types of trial:

Heavenly justice, being open to all, individuals of any rank could claim the right to ordeal by battle, though in practice it became increasingly a prerogative of the upper classes, accustomed to the use of their weapons. Fights between plebians did take place, and no doubt afforded their betters much amusement. At Valenciennes in 1455 two townsmen fought in the market-place with wooden clubs, before the Duke of Burgundy and a large gathering. The loser was promptly executed.

-- The Duel in European History, V.G. Kiernan

Talhoffer even describes a method of trial to be fought between man and wife, using a heavy stone swung in a cloth. (To balance the difference in strength, the husband is obligated to fight from a hole in the ground, which comes to his waist.) There is a bizarre story of a trial by combat between one Chevalier Maquer and a dog, which occurred in 1400. As with the Talhoffer trial, the Chevalier was buried to his waist to even the affair, which was won by the dog. But the low station of the dog in this odd story must also be understood in the context that the dog belonged to another Chevalier who had been murdered by Maquer. So it was still an aristocratic affair in the end.

Feudal convention allowed a defendant to accuse one of his judges of false judgement, and offer to settle the matter by combat. But since you were judged by your peers, this would generally only lead to knight-vs-knight or peasant-vs-peasant combats.

The limitations and potential for abuse of medieval justice traditions were understood at the time, but they were making the best of what they had. Private violence, blood feuds, and vendettas were rampant. In that context, the idea of a trial by combat that could compartmentalize the violence and prevent it from spilling out into an extended series of lawless reprials, was itself a humane advance, even when it had the immediate effect of rewarding or punishing the wrong person.

*(edited for formatting)

I wrote here a while ago on the last British example of a challenge to trial by combat which, perhaps surprisingly, dates to as recently as 1817. The challenge was made because the challenger, who was charged with rape and murder, was a burly builder who would have had an excellent chance of killing the physically unimpressive brother of the dead girl, who had brought the case against him. You might like to review the review of the case I offered at that time: it's still available here.