My textbook says 行く is instantaneous (expresses changes from one state to
another) and can only be used to describe the result of a change, not
movements that are currently in progress. But a Japanese native has told me
that it can be used to express an on-going action. For example

> 今父はコンビニにタバコを買いに **行っています** 。

describes the father somewhere in the process of the action.

So, as you can imagine, I'm very confused with all this. Could somebody please
clarify it a bit? Is my textbook wrong?
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grammar aspect verbs-of-motion
Is 行く stative, continuous or instantaneous? 「今父はタバコを買いに行っています。」
行っています doesn't express an on-going action in the same way that 走っている might be
running. The sentence means that the father has gone to buy some fags and
hasn't yet returned.
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Cited sentence obviously states a continuous state of his/her father being out
and absent. In this case continual ています is indispensable, you can never say
今父はコンビニにタバコを買いに行きます.

In instantaneous sense, like "Tomorrow my father will go to the convenience
store to buy cigarettes" (though this expample sounds pretty unreal), you can
say 明日父はコンビニにタバコを買いに行きます.

So, roughly, 行きます is instantaneous and 行っています is continuous. Please learn
about the continual ている/ています. Here are my try of showing some continous
examples similar to your one:

  * 父は出ています。 My father is out now.
  * 彼は入院しています。 He is in the hospital.
  * 田中さんは田舎に帰っています。 Tanaka-san is back in his/her hometown.

Compare these with instantaneous 出ます, 入院します, 帰ります.
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行っている often means "he has been there" but that doesn't mean you always can't
interpret it as "he is going". So, the textbook is wrong in that aspect.

  1. [habitual action] ここのところ医者に行っている: I'm going to the doctor these days 
  2. [on-going action] うまく行っている: It's going well

Edit: When you find your father walking the street and you assume
「おっ、タバコ買いに行ってるな」, that 行っている is ~~the durative meaning~~ the on-going meaning.

<https://www.youtube.com/watch?v=6jQn0ytqSm4#t=0m07s> #13 and #5 in the blue
shirt pressing the DF is what I'd yell 「行ってる！」.
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Whether something is durative or instantaneous isn't a property of verbs, but
of _predicates_ :

  * 「道を行く」 is durative
  * 「うまく行く」 is durative (as @user4092 nicely pointed out)
  * 「〇〇を買いに行く」 is instantaneous

With a **durative predicate** , you get these interpretations:

> **progressive**  
>  仕事は（今）うまく行っている  
>  "My job is going well (currently)."
>
> ｓｔａｔｅ  
>  ／ ＼  
>  **ーーーー｜｜｜｜｜｜｜？？？？？？？？？？＞**  
>  ＾ ＾  
>  ｂｅｇｉｎ ｎｏｗ  
>  ｇｏｉｎｇ ｗｅｌｌ
>
> **habitual**  
>  仕事は（毎日）うまく行っている  
>  "My job is going well (every day)."
>
> ＿＿＿＿＿＿＿ｓｔａｔｅ＿＿＿＿＿＿＿  
>  ／ ＼  
>  **ーー｜｜ーー｜｜ーー｜｜ーー｜｜ーー｜｜ー＞**  
>  ＼ ／＼ ／＼ ／＼ ／＼ ／  
>  ｄ１ ｄ２ ｄ３ ｄ４ ｄ５  
>  note: this habitual makes it explicit you are currently in the state of
> this action happening habitually, while the plain-form habitual does not
> make that explicit
>
> **perfective**  
>  仕事は（今までに二回）うまく行っている  
>  "My job has gone well (twice up to this point)."
>
> ＿＿＿＿＿＿＿ｓｔａｔｅ＿＿＿＿＿＿＿  
>  ／ ＼  
>  **ーー｜｜｜ー｜｜｜ーー？？？？？？？？？？＞**  
>  ＼ ／ ＼ ／ ＾  
>  １ ２ ｎｏｗ

With an **instantaneous predicate** , you get these interpretations:

> **resultative**  
>  父は（今）タバコを買いに行っている  
>  "My father is out to buy cigarettes (currently)."
>
> ｓｔａｔｅ  
>  ／ ＼  
>  **ーーーーｘ｜｜｜｜｜｜？？？？？？？？？？＞**  
>  ＾ ＾  
>  ｇｏ ｏｕｔ ｎｏｗ
>
> **habitual**  
>  父は（毎日）タバコを買いに行っている  
>  "My father is going out to buy cigarettes (every day)."
>
> ＿＿＿＿＿＿＿ｓｔａｔｅ＿＿＿＿＿＿＿  
>  ／ ＼  
>  **ーーｘ｜ーーｘ｜ーーｘ｜ーーｘ｜ーーｘ｜ー＞**  
>  ＼ ／＼ ／＼ ／＼ ／＼ ／  
>  ｄ１ ｄ２ ｄ３ ｄ４ ｄ５
>
> **perfective**  
>  父は（今までに二回）タバコを買いに行っている  
>  "My father has gone out to buy cigarettes (twice up to this point)."
>
> ＿＿＿＿＿＿＿ｓｔａｔｅ＿＿＿＿＿＿＿  
>  ／ ＼  
>  **ーーｘ｜｜ーｘ｜｜ーー？？？？？？？？？？＞**  
>  ＼ ／ ＼ ／ ＾  
>  １ ２ ｎｏｗ

Generally, people explain that the difference between durative and
instantaneous predicates is that a durative predicate gets a progressive
interpretation (in addition to the other two), while an instantaneous
predicate gets a resultative interpretation (in addition to the other two).

I agree with that explanation, but I hope by comparing the pictures of
"progressive" and "resultative", you can see how they are pretty much exactly
the same concept, it's just that it's the action which is continuing in the
"progressive" case, while it's the result continuing in the "resultative"
case.

# Your question

Your question really has to do with what that native speaker meant. The fact
is that 「タバコを買いに行く」 is instantaneous and 「父は今タバコを買いに行っている」 has a "resultative"
interpretation.

One possibility is that your native speaker source could have been saying that
「父は今タバコを買いに行っている」 can have the implication 「父は今タバコを買っている」, which is an action
父 is in the process of doing (though not necessarily -- while he went out to
shop, he may instead be stuck in traffic or lost).

Another possibility is that your source did not properly convey what they
meant, and they were only try to say what the "resultative" picture (and
"progressive" picture, in fact) says, namely that the subject is in some on-
going state.
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