Ward Rituals are often there as an alert system and to make it a bit harder for astral entities to infiltrate. If you attack it then alarms will go off (so this is typically not the approach you want to go for).
Note that a mana barrier spell cast on the physical plane only make it harder to cast spells through it on the physical plane (not on the astral plane), it does not impose visual modifiers for an astral observer nor does it stop wholly astral entities from moving through the location of the barrier.
...so I am assuming that the mana barrier you are talking about in this thread is a mana barrier spell cast on the astral plane, a magical lodge, a ward ritual or a circle of protection ritual.
Besides attacking the barrier you also have the option to press through the barrier or move around the barrier (but this seem to be out of scope for now). Since you are interested in breaking down mana barriers I should probably still mention that when you walk around on the physical plane you are often unaware of the existence of barriers (as they are invisible on the physical plane) and you can also normally just walk through them (or ride an elevator through them) without noticing them at all, however, if you do this while having active magic (such as activated foci or sustained or quickened spells etc) then your active magic will forcefully collide with the barrier (which will cause disruption in either the barrier or the active magic that was forced into the barrier, or all of them):
1. Activate foci and walk through the barrier.
2. Repeat until the barrier dissipate.
3.

??
4. Profit
It doesn't say if weapon foci, killing hands and spells are 'specialized' weapons to bring down mana barriers or not. Either you rule that they are or you rule that they are not. This is something you probably need to discuss at the table.
Note that, unlike previous edition, you don't actually perform an actual attack (in previous edition net hits on the attack would let you make a bigger hole). Which is a reason why you perhaps want to rule that they count as 'specialized' when it comes to attacking mana barriers.
But also note that there is no mentioning about them acting as 'specialized' when attacking mana barriers and there is no mentioning about if you should roll Charisma (which is used to Press through mana barriers) or Tradition Attribute (which is used for astral combat damage) to boost the damage value of your 'specialized' weapon. Which are two reasons why you should perhaps want to rule that they does not count as 'specialized' when it comes to attacking mana barriers.
Not sure if 'size' have any relevance on the astral plane. Either you rule that it does (and it will be hard to create a hole big enough to escape through before it recharge at the end of the combat turn) or you rule that it doesn't (and it will be easy to escape through a mana barrier as you just need any sized hole). Or you rule that the hole need to meet a specific diameter before you can squeeze through it.
Note that entities that have 3D6 initiative dice (like spirits you summon) can potentially attack twice in the same combat round.
I see nothing about mana barriers collapsing if their structure reach zero (like physical barriers that are affected by physical properties such as gravity etc). It seem as if they will just regenerate at the end of the combat turn, but I can see good arguments either way.
Mana barriers spells cast on either the physical plane or the astral plane are regular sustained spells and like all other sustained spells they can be dispelled, following standard rules for dispelling (and you are not trying to dispel something on the other side of the barrier).
Circle of Protection Ritual and Ward Ritual are both rituals, not spells - so it doesn't seem as if you can dispel them. They are both Anchored rituals. If you manage to move the physical object they are anchored to the ritual will end.
Medicine Lodges are not sustained spells and can not be simply dispelled.
If you don't bring along any astral form...
You yourself get an astral form whenever you use astral perception or astral projection (which is the reason why you can't pass through mana barriers when using astral perception or astral projection but you can just walk through them as normal when you use your physical perception)