They would need a network of Portkeys, requiring a lot of inter-Ministry support.
If werewolves were to keep traveling to avoid the nighttime in any country, they'd need a network of Portkeys spanning the entire globe. Setting up such a network would require the cooperation of various Ministries of Magic around the world: it's unlikely that one single Ministry of Magic could set something like this up for the werewolves in their country without the approval of others.
In addition, the Ministry of Magic in Britain is prejudiced against werewolves, so at least as the Ministry is over the course of the series, it's unlikely they'd be willing to make the effort to attempt to set up a global network of Portkeys for werewolves to avoid the nighttime.
Also, it might not work - it's unclear whether werewolves only transform at night.
There isn't anything indicating whether werewolves transform only at night, or if once the moon is full, they remain transformed until the moon begins to wane again. Traveling around the world would only work if the transformation was triggered every night by the sunset and subsequent rising of the moon. If, however, once it's a full moon they remain transformed until the phase of the moon changes, then they'd still transform wherever they are.
Lupin mentions disappearing once a month, but it isn't clear whether his absence during the day is because he's still transformed into a werewolf, or because having transformed the night before left him feeling weak and needing to rest.
What little evidence there is seems to imply that the transformation lasts, night and day, until the moon wanes again.
“As long as I take it in the week preceding the full moon, I keep my mind when I transform … I am able to curl up in my office, a harmless wolf, and wait for the moon to wane again.” - Harry Potter and the Prisoner of Azkaban, Chapter 18 (Moony, Wormtail, Padfoot and Prongs)
In addition, every time Lupin or someone else refers to his transformation, they say it's "once a month", not "nightly", and there doesn't seem to be any evidence he turns back human during the day. He never mentioned doing anything as a human during the day when the moon is full, even in a weakened state.