Would it be totally wrong to make a few characters in your light novel time jump/travel whilist some other characters stay in the same period of time? How would you guys take on this challange? Would you stop writing about the characters that are still in the present and focus on the group of characters that have traveled in time?
Why would it be wrong? I mean there are many stories which fulfill that scenario and they are good. You can even bound your story to the fact they are separating. "Cloud Atlas" has a very complex time lapse descriptions and it is still good and even it is built on that fact. There is no correct answer if you should stop writing for the characters which are left behind but you could also check out quest stories where a band of character take on a long way. If you think the concept is the same no matter if you travel in time or in distance. If keeping the reader informed about the characters left behind serves the story then do it, if they have nothing to tell then just leave them for now.
How you handle this is entirely up to you. As a reader, I would both want to know about the brave new world your time travelers are facing but I'd also want to know how those left behind are coping although a lot of this is dependent on how the time travel occurred. Was it intentional or an accident? Is there anything those left behind can do to "help" those in the future? If they're just wandering around wondering if the time travel worked then that's not very interesting.
If they have a purpose to still be in the story then keep them there. If they don't, then take them out.
Worst case scenario for you would be to write the parts of the people left behind the scrap it because you've decided it's not working.