For instance, "He is eating" is "Kare wa tabete iru". However, "He is dying" is not "Kare wa shinde iru". Another example is "He is going to Japan" is not "Kare wa nihon ni itte iru". So if I can't use the "te iru" form to express an action currently in progress for verbs like shinu, iku, kuru, then what would be correct way to make those sentences?
As @Hyperworm suggested, it might be good to have a more comprehensive set of examples, so I've sifted through the many sentences in the Tatoeba database. Using a python script I reduced the number of sentence pairs down to around 800 sentences, I then went through and documented the differences in around 150 of them which I thought might be relevant.
These are mainly words/expressions which haven't got
This answer is community wiki, so if there are any errors or you want to add something etc then feel free to edit it.
Words/expressions I was uncertain about but may need to be elaborated on:
As I've answered in other posts (like this one),
There's not a solid grammar-based way to express this with stateful verbs such as the ones in your example, at least as far as I'm aware. You'll probably need to use an adverb. Two such examples are the general
Either example should at least get your point across with your "He is going to Japan" example (