Asylum is based on braod UN mandate and most of the democratic countries are signatory, so most countries are obliged to provide to give refuge if applicant prove the fear beyond doubt. As like eth else countries try to make it harder and mostly they do not want flood of people. So they make asylee as second class citizen without giving them full rights. In america its different immigration is based on humanitarian approach, so they give asylee green card after one year and citizenship /passprt too. But thats not the case with other countries so 150 sth odd countries are required to provide refugee travel document. Its not like they want too. I know if any country don't want to, then it will simply say thats a diffferent story.
Heres the difference, asylum is broadly general everywhere but what to do with the displaced is based upon the host country's capacity and will. Nepal ma tetro aayeko haru bichara lai nepal sarkar ka ghusyaha neta haru le k hi garennan. And they were moved to USA. And about travelling with green card, it is only for return to america like a visa. Nothing else.
If you ask for asylum first question would be why you are asking me, but not with police of your own country. And if your government /surakshya nikayale surakshya pradhan garna sakena bhanne matra aayija yeha bhanchha. If you have own saranarthi that means you have already declared timi lai timro sarkar ma bharosa chhaina bhanera.