From Leviathon’s thread (http://learnjapanesepod.com/you-must-unlearn-all-that-you-have-learnt/) regarding ～ている to express (unfortunately) the dog dying or the dog being dead.  To say the dog is dead, we would use the ～ている form to say that it died and is remaining in that state.  But what would you say if you wanted to express that the dog is dying?  Well, since I don’t really feel like continuing that example (ワンちゃんが好きだし) I’ll use 太る(ふとる), to grow fat, instead.  (And yes this was intentional due to the similar kanji for 犬 and 太.)

Vている – Action in progress/Past event that is connected with the present
彼は太っている。 He is fat.

Vていく – Indicates a change that takes place from now on
彼は太っていく。 He will continue to get fat.

Vてくる – Indicates a change that has been taking place to now
彼は太ってきた。 He has gotten fat./He is getting fat.

The rules I mentioned in the other post also apply.  If you can express the verb with a duration, then you are talking about an action in progress, and cannot use the ていく・てくる forms to express a change.  Instead they take on the standard go and come definitions.  For example:

一時間買い物しました。  I shopped for an hour.  (Grammatically correct using a duration)
買い物してきました。 I went shopping and came back.  (Not expressing a change)

一時間分かりました。 I understood for an hour. (Grammatically incorrect using a duration)
分かってきました。 I came to understand.  (Expressing the change from not understanding to understanding)

My notes from Levithon’s post:

As for the ている form, I whipped out my Genki Vol 1 to make sure I get this correct. There are three main types:

1. Describe a continuous states.
2. Describe activities that last for some time.
3. Describe the result of a change.

For example, when you get married you go from being single to being married. One tip the book said to determine if it is talking about a continuing action or a state is if you can describe the phrase using 一時間.

〇私はきのう一時間本を読みました。
×私は一時間死にました。(”I died for an hour”, which is as ungrammatical in both English as it is in Japanese.)

So 死ぬ (Along with 起きる、行く、帰る、来る、分かる、出かける、乗る、座る、消す、忘れる、借りる、降りる、持ってくる、連れてくる、結婚する、太る、やせる and 着る) are type 3 verbs.

This is another very common spoken grammar point.  Please feel free to ask questions and I’ll try to answer or give feedback if you feel up to practicing this grammar point.