I'll have to let someone else answer for the middle ages, but in the late 16th and 17th centuries armor did have different weights. Armors during this period were sometimes "proofed" by firing a bullet at them and would often be referred to as "pistol proof", "arquebus proof", or "musket proof" specifically depending on their thickness. "Musket proof" armors were very uncommon and typically considered much too heavy to be practical outside of certain niche uses. Sir Roger Williams, for example suggests that during a bombardment a good soldier could be put in heavy armor and sent forward and scout the enemy walls for breaches. Otherwise he recommends that cavalry armor should be at the foreparts only "light pistol proof", which the rider's horse would then be able to support for up to ten hours. Armors like mail and brigandine weren't used as much during this period, but when they were they were typically considered lighter weight and only mentioned as flexible alternative to a cuirass for light troops.

Of the non-plate armors Humfrey Barwick considered Brigandine the best, but that it would still not protect against pistols, despite those being the weakest of the firearms. Toward the end of the 16th century military writers start recommending that musketeers/arquebusiers wear no armor at all, not even light armor like mail. William Garrard gives the explanation that even mail was heavy enough to tire the gunmen during a skirmish, and wearing mail was actually more dangerous than wearing no armor at all since a bullet would drive the metal links into the wound.

In the American colonies however, there was less need for protection against bullets and therefore a specific demand for "light armor" among militia. In some English settlements, records show that the colonists had a fairly high number of old fashioned mail shirts and coats of plate, which were considered obsolete in Europe but were being sent over for use against the American Indians.

While I'm no expert on the subject, I can safely say that the answer will differ massively based on time and place. No answer would accurately describe both 12th century France and 16th century India, for example. Could you narrow the question down to, say, a century and a country/region?

Most of these ideas go back to the original D&D rules, at least.

Short answer: No, they have the progression of weight backwards. Plate is the best for weight and mobility, as well as protection.

Having hauled reproduction armour and gotten fighters into it, and seen them fight mail against plate, I will aver that for the amount of fighter covered, mail is heavier than plate, and clumsier. Assessments that compare the weight of Henry VIII's tournament harness to a sleeveless, waist-length mail shirt should be obviously ridiculous. That can only be compared to the cuirrass alone. But one needs to note that if you wear plate, you don't need to carry a shield (you're wearing it), and that's quite a bit of weight. A shield is necessary to mail, because it has no real protection against blunt trauma, only cutting. Check the Bayeux Tapestry: some Norman warriors just carry tree branches to beat to death opponents in mail, because no one's wearing plate back then.

I'm guessing you've been told that guys in plate armour need derricks to get on their horses and can't get up on their own when they fall down. European knights frequently dismounted to reinforce the infantry when they were holding a position, because cavalry is only good on the move. (Delbrueck, History of the Art of War)

You can also cruise YouTube for people in repro armour, full plate, who can do forward rolls in it besides leap and run (Metatron comes to memory, especially as comparing plate with samurai armour). How much more flexibility or mobility do you need? There's joints everywhere your limbs bend. You don't need a joint between, say, your wrist and your elbow. See ffoulkes, The Armourer and His Craft for the use of rivet-in-slot joints on the body of plate armour to give flexibility.

ffoulkes had an inventory of the armour in the Tower of London, but that's mostly late and mostly tournament. However, when I had access to a copy, I noted that the war harness was lighter than the jousting harness.

"Ringmail" if we agree that's rings sewn inside linen, is lighter, but it's not much against spears, arrows, and other sharp points. You would be better off with jack, which is plates of metal or horn butted up against each other and sewn or riveted inside a jacket like the "ringmail." Scale armour, with its thick overlapping scales, is probably heavier than mail. It's also one of the early ones, being worn by the ancients.

("Mail" = chain armour. So scalemail = scale chain armour, and chainmail armour = chain chain armour armour, and I know it's common usage but it's like nails on a chalkboard to me!)

No one ever wants to talk about how the more "flexible" armour is sometimes a dangerous enemy. You can go to bend your arm, and that long sleeve of scale or mail bunches up in your elbow, so you can't bend it all the way.