教科書から学んだ動詞の使役形は、命令者は使役動詞の主語で「が」で表し、実際の動作者は助詞「に」で表す。例えば、僕が友達へ「ご飯を食べて」と言うと、僕が友達にご飯を食べさせる。「を」で表す部分は動作の実際の対象でありそうです。
なのに、気付いたのは、「に」の代わりに「を」で実際の動作者を表す場合もあります。例えば、back numberさんの歌詞の一句:「冬がそっと雪を降らせて」。ここ、「冬がそっと雪に降らせて」が正しいと思います。なぜなら、「降る」は自動詞で普段に「雪が降る」と言います。使役形にすると、「に」のべきだと思います。
他の例を見たこともありますが、今の頃は思い出せません。「に」と「を」は混用できるかどうか教えていただきたいです。
There are many points to note when using the causative. If you want the simple rule,
transitive verbs only take に, while intransitive verbs take に only if there is を already, if not use を. Many native speakers just simply avoid putting に twice even if it's perfectly fine (in some situations it's wrong and others it's disliked), and when they work that logic in reverse, they believe に is wrong if を doesn't exist...
For the detailed rule, we'll need examples, and a cup of coffee...
僕が友達（☓を・◯に）ご飯を食べさせる
僕が友達（☓を・◯に）食べさせる
A transitive verb, so you must use に.
親が子供（◯を・◯に）遊ばせる
An intransitive verb, both are correct, The difference between the two is を is permissive while に is compulsive. You can notice the difference in the negated sentences, "子供を遊ばせない" would translate to "won't let kids play" while "子供に遊ばせない" would be "won't tell kids to play" or "won't make kids play".
Naturally, you'd find native speakers mostly use を and might even say に is wrong.
子供を（◯を・☓に）死なせる
When there's a 結果的に nuance, only を is usable.
What about emotion verbs?
僕が喧嘩して、友達（◯を・☓に）泣かせた
Emotions aren't controlled by the will so I can't force my will on my friend to make him cry and hence can't take に.
Inanimate causers?
冬がそっと雪（◯を・☓に）降らせて
The winter can't force its will on the sky to make it snow because the winter is inanimate and causative doesn't like inanimate causers. So instead of being able to use both we can only use を here.
Had it been an animate causer,
先生がペン（◯を・☓に）落とす
Unlike 食べる which is transitive by nature, 落とす is trully a causative. We'll treat this is as a normal non-causative conjugation of a transitive verb so we'll use を. The reasoning is that the verb it conjugated from 落ちる(落つ) is a 非意志動詞, same as 降る.
We've seen verbs which take を for their objects, what about に verbs?
警察が犬（◯を・▽に）犯人に飛び掛からせる
警察が犬（◯を・◯に）飛び掛からせる
私が友達（◯を・▽に）家に住ませる
私が友達（◯を・◯に）住ませる
the simple rule? doubling に is bad so use を. Actually, に is reserved for a specific usage with these verbs and don't like doubling in this case. If it didn't have に in the original sentence before the causative you could've used it just fine. and the difference would be similar to 子供（を・に）遊ばせる. And you'll find natives saying に is wrong in that case because intransitive take を but they are incorrect.
使われるシーンによって異なります。
➀私《に》行かせてください。 ②私《を》行かせてください。
誰かが行かなくてはいけないときに、私にぜひ行かせてもらいたい、と主張・志願する場合は、①または②。
また②は、 私が行こうとしているとき、誰かが行かせまいとしているような状況で、私が行くのを止めないでください、という表現。 にも使えます。
Courtesy of 快仁21面相さん on this chiebukuro question.
What about verbs that take を for locations?
先生が生徒（☓を・◯に）トラックを走らせる
先生が生徒（▽を・◯に）走らせる
Can't use を twice, hence に. I haven't seen anything concrete for if を would be acceptable If you remove トラックを or just plain wrong. But ※I know natives persist that に is wrong when they don't see トラックを. So I assume both are acceptable and the difference is allowance vs compulsion?
※You'll find foreigners believing に is allowance while を is compulsion in that link, I personally found sources made for teaching foreigners which say this. It's plain wrong.
For the last one, a bonus!
先生が生徒（☓を・◯に）発言させた
上司が部下（☓を・◯に）答えさせた
Verbs where you do stuff towards people like speaking take に. Luckily most native speakers agree on this and don't use を. consider this the special rule to the simple rule.
Well that took a couple of hours... and I didn't make coffee...
