Jump and Gravity without RigidBody

How to make realistic gravity and a jump without using RigidBody and CharacterController, in the examples I see that the gravity constant of 9.81 is often used, but in other games I see that the jump smoothly moves like a wave from the top to the bottom, and ordinary gravity begins to attract gradually, that is, not sharply down, but as if gradually gaining gravity, but no matter how hard I try to reproduce it, nothing works...
How to do this?

Time to learn how gravity works then ;). Gravity is an acceleration. Acceleration is a change in velocity over time, and velocity is a change in position over time.

So when characters jump, they move smoothly up and then smoothly down not because they "gradually gain gravity", but because they gradually gain velocity due to gravity. In pseudocode:

character.velocity += gravity * deltaTime;
character.position += character.velocity * deltaTime;

I probably had more problems with the fact that the “invisible capsule (fixed position)” moved in one temporary update and the animation, of course, in a regular update and it was difficult to synchronize it all to make it look realistic...

but at the moment I think it turned out to be something more or less what was needed...