if you get a job it is of no relevance if there is a trial period or not. Also in Denmark it is normal to have trial period of 3 months.
And it is of no relevance of the job is in Denmark or not. But for job in another country you need to be sure you have documentation for the time being employed and also your income for the period.
And, in fact it will be good for you to take a job no matter if you will not not be able to continue after the 3-6 months trial period.
Because when you are on the "graduate" rate for unemployment benefits, there is a rule saying that after 6 months you can get a new (higher) calculation of you benefits rate, IF you have been working for at least 3 months. This means that you must have been working for 320 hours - this equals 24,67 hours pr. week.
So to answer you question:
Yes your unemployment benefits will stop when you start a job. And if you can't continue job after trial period, you just contact the A-kasse again.
And in your case it is important that you tell that you has received benefits as a graduate, and now (after at least 3 months of work, e.g. trial period) want a new calculation of your benefits.
Another thing: you must continue to pay to A-kasse during your trial period. Ask your A-kasse if you can continue to stay member with them, or you have to be member in the unemployment scheme in the country where you will be working.
It is important that you are member - but not important in what country.
When you come back to Denmark, you will always be able to transfer your earned rights regarding Unemployment benefits to Denmark, as long as it is within EU/EEA countries