There is not a single behavior and loanwords behave differently from one another, because after all they're strictly connected to human behavior and that's quite variated even if similar across the world.
The word Loanword itself is a calque from the German Lehnwort. And even Calque is a loanword from French. Calques are another thing but let's focus on loanwords.
For example, in English the word web both refers to a spider's web, and to the web, the internet. Italian got this last word (from www which means world wide web), so we say "Ho portato la mia attività sul web" (I got my activity/shop on the web, online).
But Italian only acquired that meaning. We don't say "C'è una web nell'angolo di quel muro" (There's a web in that wall corner). For us "Web" exclusively refers to the Internet, not to the spider's web. Loanwords often behave like this, they just refer to the object/situation/concept they referred to when they were acquired and just to that.
Another example is "Corner". In English it designates a random wall/something corner, or the Corner Kick (in Soccer). In Italian if we say "È stato assegnato un corner alla squadra avversaria", we mean "the opposing team was awarded a corner kick". We won't say "Questo muro ha un corner strano", (this wall forms a weird corner).
The same happens for the opposite: English say "lasagne" because the term was acquired from the collocation "Lasagne al forno/alla bolognese", not knowing that "lasagne" is a plural for us and that the singular is "lasagna". But English acquired that particular word and uses it as standard.
Here's another important point: when loanwords are acquired, their original gender/number is lost and they are "reset" to work in the receiving language. In Italian we say "In questa cartella ci sono molti file" to mean "In this folder there are many files".
But at a certain point, it's really hard to know what word is actually a really well-integrated loanword. I bet many Italians wouldn't know that "Grattacielo" is actually a loanword from English "Skyscraper" (cielo = sky, gratta(tore) = scraper).
English and Japanese are more transparent in this case as Japanese, for example, usually adopts Katakana so you can see what words are not Japanese. The word tsunami, 津波 (tsu: harbor and nami: wave), would probably be used just to refer to the natural disaster, even if you could say "a tsunami of crime", because people will always (un)consciously link that word to those disasters and journalists will most likely prefer another word to avoid this.
In conclusion there is not a single answer that will satisfy all instances concerning loanwords as each loanword has a different story and a different origin, but luckily for most recent loanwords, it's not hard to guess their history.
This is also true when talking about their position in the new language: some sort of usurp the original word, others will increase the vocabulary (mostly because the receiving language didn't have a word for that concept), and so on.
