The most common and most likely answer to this crossword clue is the 6 letter word OUTRUN
I've seen this crossword clue in The Guardian
O
U
T
R
U
N
More clues leading to the result OUTRUN
Buddy explains!

The straight part
We know that "Go faster in the present/ beaten in the past" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case go faster in the present/ beaten in the past = outrun.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does OUTRUN mean?
OUTRUN noun- (sheepdog trials) The sheepdog's initial run towards the sheep, done in a curving motion so as not to startle them.
- To run faster than.
- To exceed or overextend.