This post is the result of work I am doing on my own retroclone-ish roleplaying game, but to make it at least semi-useful for others I'll follow the AD&D 1st edition rules instead.
Let's start with the basics. Your character earns experience points (xp) and after a certain number of xp have been collected, your character's level goes up. For a good chunk of your character's career, every new level means more hit-points, and you roll hit-dice (HD) for those. Both the kind of HD you use and the xp required to gain a level depend on your character's class.
So let's compare a fighter and a magic-user. Let's say each character has earned 100,000 xp which puts the fighter at level 7 and the magic-user at level 8. The fighter uses d10 while the magic-user uses d4 for their HD so (ignoring all other adjustments) we get these distributions for their hit-points: 38.5 (fighter) versus 20.0 (magic-user) on average. No surprises, all good.
Now let's consider an elven character who is a multi-classed fighter/magic-user instead. This character has to split xp evenly between classes, so 50,000 xp each for the fighter and magic-user classes, putting them both at level 6. The hit-points for a multi-classed character with two classes are generated by rolling both hit-dice but then dividing by two, leading to this distribution instead. (There's a small inaccuracy here because the rules specify rounding whereas the anydice.com engine truncates by default; I tried to compensate for that by rolling all the dice at once instead of dividing each and every one by two.) The shocking truth is that our fighter/magic-user is really not much of a fighter at all: 23.75 hit-points (on average) is a lot closer to the single-class magic-user (20.0) than the single-class fighter (38.5). (In fact it's worse than rolling 7d6, which is what a single-class thief would have at 50,000 xp.)
But now compare this to a human character who starts out as a fighter but then dual-classes to magic-user. To make things roughly comparable to the multi-classing example, let's say we advance to level 6 as fighter for 35,000 xp. That leaves us 65,000 xp to pour into the magic-user class which gets us level 7. So we get 6d10 and 1d4 which leads to this distribution instead. So not only does a human dual-class character get an additional magic-user level, they also end up with 35.5 hit-points on average, very close to the single-class fighter with 38.5 hit-points on average.
So multi-classing sucks. And not just a little: It sucks big time.
Also dual-classing rules. And not just a little: It rules hard core.
Now let's do something really weird: Let's pretend we had the "unified level progression" mechanic from 3rd edition but let's pick the most expensive 1st edition progression for it. (We can't use the progression from 3rd edition "as is" because they lowered the cost for gaining a level a lot compared to 1st edition. Using the most expensive progression from 1st edition means we'll be conservative.)
Turns out that with 100,000 xp all classes in the 1st edition Player's Handbook reach at least level 7, but the Paladin progression cuts it closest with 95,000 xp required. (The monk progression requires 98,000 xp but that's for 8 HD since monks start with 2 HD at level 1.)
So if we translate forward into 3rd edition again we have at most 7 levels to play with: We could build a fighter/magic-user with levels 1/6, 2/5, 3/4, 4/3, 5/2, or 6/1. Let's pick 3/4 with the magic-user level higher than the fighter level to stay close to our previous dual-class example. All that futzing around gets us this distribution which is perhaps the most shocking one so far:
It's absolutely perfect!
In terms of averages it comes in at 26.5 which is significantly higher than the 20.0 a single-class magic-user gets. It's of course also significantly lower than the 38.5 a single-class fighter gets. This is in contrast to multi-classing with 23.75 which is much too close to a single-class magic-user and dual-classing with 35.5 which is much too close to a single-class fighter.
Like many who consider themselves "old school" I didn't expect to find something great in the 3rd edition design. But it's clearly the case that unified level progression and the revised multi/dual-classing mechanic achieve the kind of balance I would expect from mixing multiple classes. Neither of the 1st edition approaches even comes close.
Which is why my own retroclone-ish roleplaying game works like 3rd edition in this regard. Stealing the best ideas from all editions, that's my kind of old school.