I'm a native English speaker and "to run" can mean "to flee" or "to escape", especially with countries. For example, if I went to Austria to escape the United States' government, I would say "I ran to Austria." This sentance implies that she is escaping something in her home country and escaping to Austria.
True that, but sometimes it does matter. There are usage differences, which we also need to learn. So for instance 'runs to Austria' can mean either running to Austria as on a jog, or fleeing to Austria as in fleeing from the US government. Can it have both meanings in German, as well? That question is not settled by just learning that the phrase is grammatically correct.
August 13 15
Currently there are hundreds of thousands of people running in the general direction of Austria. Some are being blocked and diverted along the way. Many of those who do make it will continue on to other countries or by pass it if easier paths open up. There are hundreds of thousands more indicating they are going to try to do it too.
Now that the numbers have reached hundreds of thousands world wide attention is being directed to their progress (or lack of it). But it has been going on for at least a couple of years with the numbers in the low thousands.
I doubt that this example is intended to refer to these particular events but masses of people running to or from various borders does happen, especially in Europe with its many contiguous countries.
Capital cities of Austria and Slovakia, Wien and Bratislava respectively, are only 60 km apart. One can cover this distance on foot or by bicycle. Unless one is an American that is ;) Many towns and villages are located just a few km across the border (Leibnic, Passau, Salzburg, Maribor...)
To me "runs for Austria" sounds more like she would be running in a race in the Austrian team, which has a totally different meaning than "runs to" (a location).
Even though you probably in some case could translate "nach" to "for" (Only example I could come with, a bit clumsy: "We are heading for Austria" - "Wir sind auf dem Weg nach Österreich" and even there you could also say "heading to"), it doesn't mean that they would be interchangeable.
EDIT: Look for is a much better example, thanks.
If you have doubts about when to use nach and zu I found this article very helpful https://www.thoughtco.com/say-to-in-german-nach-4069659
The audio recognition is really inaccurate. When i say something simple correctly it says i am wrong and more than once, when i completely screw up a sentence it says it is right... Once i said nothing and it said i was right, another time i accidentally translated and said the English and it said i was right and just now i completely tripped over my words and it said i was right. ???
The translation of a preposition generally depends on the meaning. E.g. nach would translate to after in the context of time or order, to for destinations of a trip etc, or even in for constructions like „meiner Meinung nach“ – “in my opinion”.
The to translation for destinations with the destination being a city or a country is usually nach. Exceptions are countries that require an article (the minority), which are generally countries in the plural, masculine and feminine countries. They require an in + article. I can't think of any country that is neuter but still requires an article in general.
„in den Irak“ (Iraq)
„in die Schweiz“ (Switzerland)
„In die USA“
In sentences such as Ich will nach Frankreich or Ich muss nach Hause, the "go" meaning doesn't come from the nach but, if anything, from the will or muss.
But probably not from any one verb but from the whole sentence where, by convention, it's clear that the missing verb is one of movement.
A bit like in English where if you say, "Right you guys, home now", you know that the missing verb is "go", but not because "home" can mean "go home", more because you can leave some things out of some sentences and the listener will "fill in the gaps".
"rennen" cannot mean "to have control over", so "rennt :Österreich" doesn't make any sense as "control Australia. The woman is running to Austria. "Nach" is literally "to" here. If you're going to the store or some location around the city, you use "zu". If you're going to a country or your house you use "nach".
At first I was like "whoa. what? That's crazy!" Then I remembered that European countries are really small and close to each other and don't have crazy strict border regulations. I always forget that you can make a day trip of going to Austria and back (from Germany), however in America you can't even cross a singular state in a reasonable amount of time...
My understanding is that "zu" (a dative preposition) means at, or to and is used with people. "Nach" (a dative preposition) means to, or to go to and is used for countries, cities, etc. (exception is feminine countries). Also, by train and right, left, north, south, east, west. nach Hause = homeward; not at home
You said nach is not used with feminine countries -- and indeed, we say ich fliege nach Frankreich but not ich fliege nach (der) Schweiz.
Instead, we use the preposition in with non-neuter countries: ich fliege in die Schweiz, die Türkei, die Slowakei; in den Iran, den Libanon, den Kongo.
"means" is a tricky word.
Let's just say that when travelling to a non-neuter country (masculine, feminine, or plural) is intended, German usually uses the preposition in while English would use the preposition "to".
(Plural examples: in die Niederlande, in die Vereinigten Staaten.)
And the verb "go" may sometimes be left out with this preposition just as with other prepositions indicating the destination of a journey -- e.g. ich muss zur Bank / in die Schule / nach Frankreich / in die Schweiz for "I have to [go] to the bank / to the school / to France / to Switzerland".
So nach means "after" AND "to"?
I don't think "means" is a good way at looking at things. German is not a code for English, where you just replace an English word with a German word, always the same German word for the same English word.
That works all right for some things, e.g. Katze means "cat", but prepositions are notoriously different across languages. (Even inside English: Why do we "listen to somebody" but "look at somebody"? Does "to" mean "at"?)
So nach can be translated as "after" in some instances and as "to" in other instances. And zu can be translated as "to" in some instances. Conversely, English "to" can be translated as "nach" in some cases, "zu" in others, not at all in others (e.g. indirect objects, which are simply in the dative case in German, without a preposition).
So why not "... rennt zu Österreich"?
Because Austria is a country, not a place such as a bank. We use nach with cities and countries.
Why can't this be translated as 'The woman runs towards Austria',
Because that's not what the German sentence means.
"towards" means in the direction of, but does not say that she reached that destination. German nach Österreich does mean that she went all the way to Austria, not merely towards it (= Richtung Österreich).