keep up with

verb3

1. •
   To manage to remain beside or just behind someone or something that is moving away from one.

   • Blake needed to run to keep up with the indignant Doctor. — 1993
2. •
   To manage to follow an argument, a discussion, etc.
3. •
   (figurative) To manage to remain up to date with trends, fashions, etc.

   • Keeping up with the latest hair trends as ever, Wella has just introduced Hair Sets in three variants. — 1992