He seemed to speed up in easy places and slow down in more difficult ones. Easy places for him meant ascending and descending sequences and difficult ones - around cadences.
I think it is logical that sequences are easier than cadences because in sequences, the music simply repeats in predetermined manner up or down. In other words, composer uses the same melodic and rhythmic idea but transposes (more or less) it from different pitches.
Sometimes the sequence modulates to another key and sometimes it stays within the same key. At any rate, since there is only one idea, once you learn how to start it, the rest of it continues in a fairly straightforward and predictable manner.
The cadence can be much more difficult to play because often the harmonic rhythm changes faster than anywhere else in the piece. In other words, around cadences there is too much going on musically and so the challenge is to play it in the same tempo.
Now this student appearently felt quite strong during sequences and less so around cadences. Therefore his tempo fluctuated. It was especially noticeable in descending sequences when he began to play faster and faster as if he was in a race.
Have you been in such situation yourself? If so, I think the best solution always is to count out loud the beats of the measure. When you are keeping track of the pulse, saying the numbers of the beats out loud prevents from speeding up or slowing down because it's too obvious.
Of course, if you can't play fast enough, slow down and choose a tempo while checking the most difficult passages of your piece first. Sometimes it means you have to master them on a higher level and repeat many more times than the easier places in your composition.