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Can I think of the participles sentences below like these?
1. I saw him reading a book. => I saw him (who was) reading a book.
2. I found a dog killed on the road. => I found a dog (that was) killed on the road.
3. A dog came running. => A dog (that was running) came.
4. The injured man was carried to the hospital. => The man (who was) injured was carried to the hospital.
cf) what's difference between "carried to the hospital" and "taken to the hospital?"
5. He went out with his dog following behind. => He went out with his dog (that was) following behind.
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Comments  
I don't think that is a good idea for some of them; it does not create the same conceptual relationship:

1. I saw him reading a book. => I saw him (while he was) reading a book.
2. I found a dog killed on the road. => I found a dog (that was) killed on the road.-- OK.
3. A dog came running. => A dog came (and he was) running.
4. The injured man was carried to the hospital. => The man (who was) injured was carried to the hospital.-- OK
cf) what's difference between "carried to the hospital" and "taken to the hospital?"-- Nothing, really, unless he walked there with a friend.
5. He went out with his dog following behind. => He went out and his dog (was) following behind.
moon7296 what's difference between "carried to the hospital" and "taken to the hospital?"
1. Carried to the hospital
CARRIED TO HOSPITAL


2. Taken to the hospital

Being taken to the hospital
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Please help me with this.

I have a doubt with this one.

5. He went out with his dog following behind. => He went out and his dog (was) following behind.
The use of behind sounds somewhat odd to me here for following itself implies that.
Nevertheless, it is a common collocation. Note that the dog could also follow beside him, another common collocation, Suresh.
Thank you, Mister Micawber.
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Mister Micawber5. He went out with his dog following behind. => He went out and his dog (was) following behind.
Q1) Do you know why exactly "He went out with his dog following behind."(#1) is used instead of "He went out and his dog (was) following behind.?"(#2)
Is it simply because the economy of reduction? or to deliever a subtle difference from #2 to #1?

Here's another example of the similar structure.

One of the main Indian areas for carpet manufacturing is Uttar Pradesh. Children there live and work in the factories, sleeping at night on the floor of the same huts where they work during the day.

Q2) The reason why this participle structure used is the same reason as #1 and #2?
moon72961) Do you know why exactly "He went out with his dog following behind."(#1) is used instead of "He went out and his dog (was) following behind.?"(#2)Is it simply because the economy of reduction?
Yes.
moon7296or to deliever a subtle difference from #2 to #1?
I see none such.
moon7296One of the main Indian areas for carpet manufacturing is Uttar Pradesh. Children there live and work in the factories, sleeping at night on the floor of the same huts where they work during the day.Q2) The reason why this participle structure used is the same reason as #1 and #2?
Yes, I would consider it a more succinct style, that is all.
"I saw him reading a book. " which tense is this?
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