Projectile Clashing
The way Projectile Clashing works in Marvel Vs. Capcom 3 is that anything that counts as a Projectile is assigned "Durability Points", basically a point value indicating how strong it is as a Projectile. Thus, when two Projectiles clash, the Projectile with higher durability will win and pass through the other Projectile, but the winning Projectile's durability will have been reduced by the durability of the Projectile it beat and, thus, can start losing to weaker and weaker Projectiles.
This is why you'll see in a lot of cases it takes a few shots of something to finally penetrate another Projectile. Amaterasu's Ice Beads, for example, have 1 durability point per shot, so it will five shots to finally eradicate Ryu's Hadoken, which has 5 durability points.
All projectiles move differently, but can be categorized as a moving projectile, stationary projectile or beam.
Moves like Ryu's Hadoken travels across the screen slowly, the entire projectile with 5 points of durability.
Moves like Dr. Doom's Plasma Beam are beams, will travel across the screen very fast, and last 12 frames with 1 durability point per frame. (You can think of a 12 frame beam as being 12 separate fireballs firing every frame.) So if the Hadoken is midscreen, Plasma Beam will consume the fireball, and continue to hit Ryu with the remaining hits. However, if the Hadoken is 4 frames from hitting Dr. Doom, Plasma Beam will not beat the fireball in time, and Dr. Doom will be interrupted.
Moves like Dormammu's Black Hole are stationary projectiles; they appear instantly one place and will stop incoming projectiles and beams until the durability or duration runs out. Because stationary projectiles have no travel time, they can beat beam projectiles by interrupting the opponent.
A special case of stationary projectiles is Storm's Whirlwind, which looks like a moving projectile but is actually several stationary tornadoes! These stationary tornadoes get created in timed sequence regardless if the previous tornado was destroyed, (as long as Storm isn't hit.) So if Dormammu does Chaotic Flame, which beats all normal moving projectiles, Heavy Whirlwind can still hit Dormammu since whirlwind is not a moving projectile! (To test this out, have Storm cast Heavy Whirlwind from full screen and then X-Factor and block. Then have Dormammu cast Chaotic Flame and then see the first few tornadoes be snuffed out, yet the final few tornadoes still get created and hit Dormammu!)
Beam supers have odd behavior and don't clash with consistent rules as normal beams and projectiles do. Super projectiles (like Akuma's Gohadou Agyo or Sentinel's HSF) don't clash with beams, and instead go through each other! (Meaning Proton Cannon won't stop any fireballs from a Gohado Agyo. Likewise, you won't get off the full Gohado Agyo because the beam will still interrupt you halfway.) Also, certain beam supers don't clash and instead pass through each other! so whoever did the super first or has the faster beam, will hit first. (Or if done at the same time, will trade hits! i.e. Akuma's Gohado Ungyo vs Akuma's Gohado Ungyo) Some beam supers do clash fully (Iron Man's Proton Cannon vs Iron Man's Proton Cannon), and others only clash partially (Iron Man's Proton Cannon vs Akuma's Gohado Ungyo). More research will need to be done to explain why some beams clash and others do not.
Priority
There is a school of thought that all projectiles can be categorized as low, medium, and high priority projectiles. (Alternatively, you can think of medium priority projectiles as having massive amount of durability points, ie 1,000 points; and high priority projectiles as having even greater durability points, i.e. 1,000,000 points; so that one type always beats another.)
Most projectiles are low priority, and will interact with other normal projectiles based on durability points as stated above. Beam projectiles usually have 1 durability points per frame.
All supers projectiles have high priority, and will interact with other supers same as above. However, Super projectiles will always beat normal projectiles, both low and medium priority ones. So Trish's Round Harvest super will beat Dr. Doom's Plasma Beam.
Some special projectiles have medium priority.
These are special moves that will always beat low priority projectiles. For example, Arthur's Lance projectile powered up with Gold Armor is medium priority, and will always beat Dr. Doom's H Plasma Beam. Super projectiles will still always beat medium priority projectiles.
Some special and even normal moves will nullify projectiles completely during its active frames. For example, almost all of Dormammu normal attacks will nullify normal projectiles (and beams too) regardless of durability. Another example is Amaterasu's green shield, which will nullify normal beams and reflect normal projectiles!