Based on the numbers given in the arena for speed, as well as my experience from adventure mode, a full suit (helm, breastplate, mail shirt, greaves, gauntlets and high boots) of steel or iron armor does not encumber a dwarf at all.
This is only true for dwarves that aren't weak. I have a dwarf that is described as "very weak", his strength value in Therapist is 519/1731 and he's quite slow even though he's only wearing a bronze mail shirt, wooden shield and a bronze crossbow. At the same time, I have two dwarves that are "very strong" and they're running around in steel set, with breastplate and mail shirt, and their speed is normal. They're all dabbling armor users.
Sorry, clearly my previous post was too long, and you didn't read it. Here's the relevant bits:
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The only way you would have an issue is if you use copper/bronze armor, or with marksdwarves, as bolts are stupidly heavy.
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The only potential issue is that both fortress and arena-created dwarves have some variation in size (and hence carrying capacity), but assuming you train your dwarves at all they will be above average size.
Since you are doing both of these things, this is why you are having a problem. Also being very weak (weak = small size) could be enough to slow down a dwarf somewhat using the steel armor I described, but melee military training will build up size/strength up to normal levels within a few months.
I do not recommend full steel for marksdwarves, as combined with their bolts this is too much weight. I would give them a steel helm, boots, and gauntlets (as these are all light), leather armor and bone greaves. Leather and bone armor won't protect against metal weapons, but will do something against animals, punches, etc. Also, armor doesn't wear out, so then you don't have to make clothes for these dwarves. I tend to recruit all my civilian dwarves as marksdwarves, and assign them bone helm, gauntlets and greaves, leather armor and boots (until metal becomes available), as this armor is better than nothing, and then I don't have to deal with worn out clothes.
Armor user decreases the weight of worn armor linearly with skill level, with full weight at dabbling and zero weight at legendary. However by the time a dwarf gets to legendary armor user they will also have incredible size and strength due to training, so wouldn't be impaired by armor weight anyway. Armor user is really only meaningful in adventure mode, where you are wearing armor and carrying tons of stuff around, or if perhaps you want your legendary military dwarves to haul rocks for some reason...