Honestly any player can move between leagues and play well all the time, just everyone needs to adapt, I think it's obvious. But yes, there are some players who can't play in a new league right from the start, but in a new team in principle, I think it's much easier to adapt to a team than to a league
There are definitely also players out there that are having a pretty hard time to adapt whenever they are joining a new team or even a new team in a new league and there are several examples for that out there. Just take a look at Messi. He pretty much was the dominating player of football (together with Cristiano Ronaldo) in the last 10 years or so when he still was at Barca. Now at Paris Saint Germain he is still a great player but definitely not the same as he was back in Spain. Age may be one factor but that's only a part of the story. I think he is a player that just needs to fell comfortable to perform at his highes level.
you gave a good example of messi, not all players get used to it when they go from one league to another. messi at PSG is not being the same messi and that is not because of his age and precisely because of the style of play of PSG and france, I believe that for example if messi had gone to a spain team like real madrid or atletico madrid he would continue to play well and scoring a lot of goals and you wouldn't notice any change, but in france or even if he had gone to england or even italy he would have had a very average performance, in the case of cristiano ronaldo is the type of player you get used to in any league, but that's because he started in portugal where he has the same style of play with france, england, germany and italy