So my suggestion is to make the base cap rate (in Assault matches) proportional to the % of Mechs left in the battle (counting all teams).*
i.e. if most people are still alive, then the cap rate should be pointlessly slow. Once you start getting towards 50% of Mechs destroyed, the cap rate rapidly increases to around what it is now. I'd suggest it then stablises (as it doesn't really need to keep getting faster). In other words, it would be an S-curve. Though the precise formula and relationship would obviously be worked out with testing (the point here is the concept, not the maths).
This will hopefully encourage teams to actually seek combat. If you get to the enemy base without finding any of them, well just keep moving!
Notes:
* Using % works better than a pure number as it automatically adapts to battles with different numbers of players (e.g. 12v12, 8v5, 12v10, or instances where a team is short).
I'd initially considered that it should be based on just the % of enemy Mechs remaining, but figured this unnecessarily disadvantaged a team that's lost more Mechs than their opponents (i.e. since your opponents have more Mechs left, they can already cap at a faster rate than you without your rate being slower too). It also avoids situations where teams have a different cap rate because they started with a different number of Mechs (1/8 != 1/7).