Gipphe answers it best: pushes and pulls are two forms of hard cc that interrupt, or more technically disable champions. Abilities that are interrupted do not finish their effects, so your jump will not do any damage if it is interrupted.
If that doesn't satisfy you, let's get into the meat of things...
What's important to differentiate here is whether the jump is a continuous movement ability or a teleport/blink. In League of Legends there is no concept of a height or being mid-air; you are always moving along the ground at the same height as everyone else. This means that you are still affected by all abilities as normal while jumping. During continuous movement your hitbox travels with you in a straight line along the ground as you jump. This is why you can still be hit by Veigar's Event Horizon while jumping: your hitbox moves right through it! During blinks your hitbox instantaneously teleports to another location, so you cannot be hit by abilities on your way. This is why blinks like Flash do not trigger Veigar's Event Horizon - your hitbox never passes through it.
Pushes and pulls get special treatment in terms of continuous movement abilities because you can't be continuously moving to two different places; something has to take precedence. What takes precedence is the last applied ability. So if Tristana Rocket Jumps and is then Rocket Grabbed by Blitzcrank, her jump will be interrupted and she will be pulled towards Blitzcrank. If she is Dredge Lined by Nautilus before Blitzcrank finishes pulling her, she will instead be pulled towards Nautilus as seen here.
Tristana's Rocket Jump would take precedence if used after Blitzcrank's Rocket Grab. Unfortunately for her, most continuous movement abilities cannot be used while you are being pushed or pulled, so you never get the chance to have your own ability take precedence. On the other hand, blinks have a varying window where they can be used after you are pulled. If they are used after then they take precedence, effectively blinking you to safety. The easiest blink to perform is Ezreal's Arcane Shift, followed by Katarina's Shunpo. Flash unfortunately gives little to no window for use after being grabbed, so it's difficult to Flash to safety before you are completely pulled in.
It's easiest to determine if your ability is continuous movement or a blink by how quickly you travel there: either over time (whether quickly or slowly), or instantaneously. Some examples of continuous movement abilities:
- Tristana's Rocket Jump
- Lee Sin's Resonating Strike and Safeguard
- Jayce's To The Skies
- Jax's Leap Strike
Some examples of blinks:
- Katarina's Shunpo
- Ezreal's Arcane Shift
- The summoner spell Flash
And there are also continuous movement abilities that cannot be interrupted. This means no form of cc can affect them. Some examples:
- Malphite's Unstoppable Force
- Vi's Assault and Battery
- Hecarim's Onslaught of Shadows
- Shyvana's Dragon's Descent
Because continuous movement abilities still move along the ground, they also trigger traps on the ground. This includes Nidalee's Bushwack, Caitlyn's Yordle Snap Trap, and Teemo's Noxious Shroom. You are immolibized by Caitlyn's trap and slowed by Teemo's Noxious Shroom after your jump.
Malphite's Unstoppable Force used to be interruptable, which was highly anticlimactic for Malphite players when they used their ultimate.
Until recently blink abilities could be interrupted/cancelled by certain flings and pushes. Theses interactions have been cleaned up recently because they were pretty silly, especially given that Flash would be cancelled and put on cooldown (harsh). See patch notes 3.8 under General Champion Notes. In particular they targeted Jayce's Thundering Blow, Singed's Fling, Skarner's Impale, and Volibear's Rolling Thunder. I'm in the process of looking for a replay of this, but I clearly recall Voyboy's Jayce performing a flash cancel and I've done it myself (pre-patch).