We can't answer that from the "what if?" angle as we don't do what-if questions here, but we can look at what did happen when nobles and monarchs gained titles abroad.

It's important to note that, despite the tendency people have to talk about "medieval Europe" as a homogeneous entity, it's basically ungeneralizable - the way a Polish monarch in 966 handled a situation may be very different from the way a French one in 1113 or a Swedish one in 1400 did. So there's no rule that dictates what a monarch would do and how the title would be inherited. This situation you're describing also didn't happen often; while we all like stories about contested or unexpected successions, most of the time, it was pretty mundane.

That being said, there are a few examples that can be said to fit under what you're discussing.

The Angevin kingdom of Naples was not really just a kingdom focused on southern Italy - its monarchs also ruled the counties of Provence and Forcalquier (what is now most of the administrative region of Provence-Alpes-Côte d'Azur), some of northwest Italy contiguous to them, the duchy of Durazzo/Durrës in Albania (directly opposite the heel of the boot of Italy), part of the Peloponnese peninsula of Greece (occupied by the Franks during the Crusades), and Sicily (until 1282). This meant that anyone who inherited the throne of Naples also inherited these other titles that weren't geographically attached to southern Italy. They also claimed the title of King of Jerusalem, as Charles I had paid Maria of Antioch, granddaughter of Isabella and Amalric of Jerusalem, for her claim to it, but technically they didn't always hold administrative control over it. All in all, the kings of Naples were extremely big power players in European politics in the High Middle Ages! I'm not sure entirely how the counties/duchies/etc. were ruled administratively, but it wasn't seen as inappropriate or problematic among monarchs for disparate regions to be ruled from afar - remember, this is before the advent of nationalism, and the people in e.g. Durrës wouldn't have been upset not to be ruled by a genuine Albania, probably. (Which also addresses your line about local nobles electing someone else - there was generally not a sense that the highest rank in the land should be of the local ethnicity. Genetically speaking, monarchs typically intermarried enough with foreigners that they were their own mixed ethnicity anyway.)

So lets say the Duke of Aquitaine marries a noble in Flanders, Bavaria or somewhere further than that. His wife eventually inherits a title because she became the only eligible heir (maybe all male siblings died, etc). So what happens next in a scenario like this? Would the wife keep or give up the title?

A good example to address this specific situation is Juana I of Castile. She was the third child/second daughter of the renowned Isabel of Castile and Fernando of Aragon, so it never seemed likely that she would actually inherit anything, making her a good pawn for a marital alliance; she was married into the Burgundian branch of the Habsburg family, while her husband's sister was sent to Spain to marry her brother, as part of an alliance between the Spanish kingdoms and the Holy Roman Empire to protect their interests in - tada! - Naples, against French incursions there. She was 16, and when she left Spain for Burgundy it was assumed she wouldn't come back. Over the next ten-ish years, she would bear children with Philip, Duke of Burgundy, and be a model duchess.

Everything changed when her brother, older sister, and nephew died, leaving her as the her parents' heir. Spain was a much bigger deal than Burgundy, so they left the children behind and went to Madrid so she could be recognized as the next ruler; eventually, Philip would go home and use their children as emotional leverage to get her to come back to his soil. It had been agreed, at Isabel's insistence, that Fernando would not inherit Castile, so when Isabel died a few years later, Juana became queen of her mother's territories and went there with her husband to stay.

There's definitely more to her story (you can read an answer of mine on her here), but that sums up the aspect of her life that relates to what you're asking about ... except that, as you've specifically asked about a wife inheriting a title, you've got to consider the struggle for power between a husband and wife. In many situations of women inheriting earlier in the Middle Ages, they simply relinquished control to their husbands, who became kings de jure uxoris, or, if they were widows, to their sons. (I discussed men ruling through their wives/mothers in this previous answer.) Fernando and Philip both arrogantly decided they should hold power over Castile, completely disregarding Isabel's will and Juana's rights, and once Fernando and Philip were both dead, her son did the same. To some extent, we can say that a woman who inherited a title lower than her husband's would hold onto it to pass it down to a child, and a woman who inherited a higher title would probably see her husband doing his utmost to take it from her if she didn't cede it to him.

I doubt dukes and kings would have liked it when some foreign noble inherited land in their country, because thats effectively a loss of territory.

For that, let's look at the Angevin kings of England and the Hundred Years War. This is a totally different branch of the Angevins from the kings of Naples! These descended from Geoffrey, Duke of Anjou, who married Empress Matilda of England (who I discuss in this prior answer ostensibly about Mary Tudor): Henry II and everyone that came after him to the end of the Hundred Years War are considered Angevins. Through Henry II's marriage to Eleanor of Aquitaine, the Angevins held quite large portions of French territory, which was a large bone of contention between England and France. Technically, the English kings were vassals of the French kings when it came to those lands, but the former didn't really consider themselves such.

When Charles IV of France died in 1328, for the second time a king of France left no son to follow him, and for the first time, he had no brothers to inherit either. In the end, as I explain in greater detail in this answer about the Capetian dynasty, it was declared that not only could no woman be a reigning monarch of France, no woman could even pass on a claim to the throne, which was against all norms in European courts, and a male cousin became the next king of France rather than either of Charles's daughters. Edward III of England was willing enough to buy that Charles's daughters couldn't inherit, but as his mother had been Charles's sister, he was a closer blood relation than the cousin who became king of France, and he pressed his own claim on that right. This was hot on the heels of a war between Edward's father and Charles IV over these English French lands that Edward II basically lost, although they kept Aquitaine (by giving it to Edward III while he was still a prince). So it's not entirely surprising that Edward III would press that claim militarily, while at the same time the new Philippe IV of France would try to take back the land that Edward personally held in his kingdom. This war over the rights of English kings to the French throne and to French lands would last for generations.

If you have more granular questions about the Hundred Years War, it would probably be best to ask them separately, as I'm sure other flairs here have a better handle on the overall politics and the details of the variously turning tides, but I will try to handle any follow-up questions you may have!