gain on

verb4

1. •
   To encroach on.

   • The ocean gains on the land.
2. •
   To win ground upon; to move faster than, as in a race or contest.

   • Soon, the van lurched into a high gear and started gaining on Zoe, who was running as fast as her little legs would carry her. — 2012
3. •
   To get the better of; to have the advantage of.

   • The English have not only gained upon the Venetians in the Levant, but have their cloth in Venice itself. — 1705
4. •
   To obtain influence with.

   • My gentleness and good behaviour had gained so far on the emperor and his court[…] that I began to conceive hopes of getting my liberty in a short time. — 1726