To keep up with somebody is to move as fast as they do. Keep up with the class, keep up with the payments, hard to keep up with.
It works for walking, for work and for news alike. Keeping up with the news and with the bills are the two commonest uses, and neither is about speed of foot.
1phrasekeep up withTo move as fast as somebody else.
“She could barely keep up with him on the hill.”
keep up with is a phrase.