The whole issue is a syntactic one.
The phenomenon is called in English "predicate noun over the object" or in the PMEG "perverba priskribo de objekto" (which in my humble opinion is an excellent way of expressing it).

Let's take sudanglo's examples:

A) Li vidis Jesuon venanta al li.
B) La disĉiploj vidis Jesuon starantan dekstre de Dio.

In A "venanta al li" is a predicate noun over the object "Jesuon" and hence it isn't assigned accusative.
In B "starantan dekstre de Dio" is an attribute (adjunct) of "Jesuon" and hence it agrees with "Jesuon" in both number and case.

The thing is, that adjuncts as in B are part of the nominal phrase they describe, while predicate nouns are phrases of their own. Maybe some parenthesis may help:

A) Li vidis (Jesuon) (venanta al li).
B) La disĉiploj vidis (Jesuon starantan dekstre de Dio).

If we now ask for the different phrases (asking for phrases is every syntacticist's hobby), we get:
A) What did he see? - Jesus.
A) And what was Jesus doing? - He was coming to him.
B) What did they see? - Jesus standing next to God.

Also these vile predicate nouns come with two toppings: "over the subject" and "over the object". And they come in two flavours: stative and resultative.

Mi vidis ĝin kaputa. (stative)
Mi farbis ĝin blua. (resultative)

Another example:
C) Mi farbis bluan vandon.
D) Mi farbis blua vandon. (More commonly: Mi farbis vandon blua, but don't bet on word order.)

C means "I painted a blue wall" and D means "I painted a wall blue."

Rules Summary:
Adjuncts agree in both number and case.
Predicate nouns over the object agree in number but not in case.
Predicate nouns over the subject agree in number and case if they are base-derived adjectives, and don't agree neither in number nor in case if they are participles (in this case use -e).

sudanglo:I believe this is discussed somewhere in PMEG.

Yes, here.

BoriQa:
Doesn't the participle act as an adjective? In which case it should take the accusative ending?

I'm no expert, but does this have anything to do with the participle acting as part of a participial phrase as opposed to being a stand-alone adjective (even in the case where there is only the participle)?

In "Mi vidis Johanon murdanta gnomojn," you have "murdanta gnomojn" as a phrase describing the object Johanon, but it is describing in a somewhat different way than a normal adjective can describe. It seems to depend on the verb, and is linked to how the subject verbs the object (it doesn't simply describe the object). For "Mi vidis Johanon kolera" I am only seeing John mad in this instance, but in "Mi vidis Johanon koleran" I am seeing "angry John". The accusative more strongly links the adjective to the object and does not depend on the verb at all (Angry John went to the store, etc.).

So "faranta tion" and "diranta tion" would be phrases. This leaves the question of when a single participle is a participial phrase or just a participle. I would say it is a simple participle if it is acting like an adjective like in "blue house" or "angry John" that always (or at least semi-permanently) describes its noun. The phrases are a much more temporary affair.

This is similar to how we use a participial adverb/adverb phrase to describe the subject (indirectly through the verb, mind you),"Murdite de Johanon la gnomoj fantomus lin la tutan vivon." The phrase is describing the subject temporarily, at the time of the verb's action.

Also, the sentence "Mi farbis la domon rugxan blua" illustrates a verb-specific property where we can describe what we do/did/etc. to the subject. I suppose it is somewhat related to the participial phrases.

I hope all this made some sense.

kaŝperanto:The accusative more strongly links the adjective to the object and does not depend on the verb at all (Angry John went to the store, etc.).

This is exactely the case.

An attributive adjective or participle is an integral part of the noun it describes. Or as you put it, it "more strongly links the adjective to the object". In this case the adjective agress in both number and case.

An predicative adjective or participle is linked to the object via the verb. This is why I think that the term "perverba priskribo" is a lot better than "predicate noun over the object". In this case it agrees in number, but not in case.

Why? The number is a semantic feature. It comes from the noun/adjective itself and changes its meaning. "dog" means something different as "dogs".
The case is a syntactic feature which the verb assigns to its object. IT doesn't add or change any meaning of the word, it is just the way verbs say: "Hands off, this girl is mine". The predicate noun isn't the object of a verb and nobody assigns case to it. The case doesn't change the meaning of a verb "hundo" and "hundon" both mean dog, the -n is just necessary to make the word fit into the clause. [1]

((I don't know if you are into computer science, but here a simile: Number is defined at compile time and case as run time. For L-Typed languages at least.))

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[1] Which doesn't mean that other languages don't assign case to predicate nouns. Latin and German verbs can assign accusative to both the direct object and its predicate noun (double accusative as in "Plebs creavit Petrum consulem" or "Ich nenne dich einen Esel." ) In Russian for instance (and I think in other slavic languages too) the verb can assign instrumental case to the predicate noun. "Он был врачом." Other language might do other wonderful things. But as far as Esperanto goes, the Legal Codex of 1898 (rev. 1902) states in Article 128 Letter B "No verb shall assign case to a predicate noun". )

Personally, I find neither "predicate noun over the object" nor "perverba priskribo de objekto" particularly transparent or helpful descriptions.

Also lumping together Li farbis la pordon ruĝa with Li vidis Jesuon venanta al li doesn't seem to me to have much explanatory usefulness.

Granted they are both cases where an object of the sentence is followed by a nominative, but they seem different, and not just because one might be thought of as resultative and the other stative. (Though this is may in itself be a useful distinction. There is clearly a difference between vidi iun kolera and igi iun i-a)

When it comes to exposing a syntactical difference between an agreeing participle and a non-agreeing participle, by re-expressing in other words, it is not clear that the 'tranformation' is going to be the same when the participle agrees or always different from cases when the participle doesn't agree.

I agree with Kasx, that there is a more adjectival force when the participle agrees in case. But there is a difference between the a standing Jesus and Jesus standing,

sergejm:Mi vidis ke li faris tion.
Ŝi aŭdis ke mi diris tion.

Please also don't forget that after vidi and other verbs of perception usually the indirect speech is used, see PMEG for a list (where, for some reasons, the verbs of perception are lacking, but a quick search in the Tekstaro shows that at least Zamenhof overwhelmingly used e.g. vidi that way).
So:

Mi vidis, ke li faras tion.
Ŝi aŭdis, ke mi diras tion.

nornen:
An predicative adjective or participle is linked to the object via the verb. This is why I think that the term "perverba priskribo" is a lot better than "predicate noun over the object". In this case it agrees in number, but not in case.

...

((I don't know if you are into computer science, but here a simile: Number is defined at compile time and case as run time. For L-Typed languages at least.))

I also find perverba priskribo to be a little more helpful, but that is because I am willfully ignorant of the finer details of grammar terminology.
That's one thing I like about Esperanto and the PMEG; the terms are usually very simple/straightforward. Instead of inventing a word such as predicate _____, we get a "by-means-of-a-verb description", which is immediately understandable by a novice grammarian.

I think I get where you are going with the analogy. As an embedded C guy I rarely use those newfangled dynamic and loosely-typed languages myself.

I saw him do it - there's some uncertainty with this as to whether you just saw him doing it or you actually observed the completion of the action. This might be picked up on in a witness's statement in a court of law.

So perhaps the best translation in Esperanto is Mi vidis lin fari tion.

Mi vidis, ke li faris tion leans more towards I saw that he had done it.

For observation of the act in progress - mi vidis, ke li faras or mi vidis lin faranta

BoriQa:

nornen:Another option would be participles:

Mi vidis lin faranta tion.
Mi auxdis lin diranta tion.

Nota bene: In this case, the participle does not take the -n ending, albeit it refers to lin which has an -n.

... now I have a doubt about the accusative on "tion".

The double accusative confuses me.

Why does it take the accusative (given that "lin" is already in the accusative)?

I'm almost sorry I asked...

Thanks everyone for all the responses, and for the time spent explaining the accusative on this example.

I now have a better understanding, and can recognize it better when I find it in writing.