Introduction tldr We have observed and described with formulas how electrically charged bodies repel each other or are attracted by bodies with electron deficiency.
We see this also in the elementary scale, when an electron is attracted by an ion with proton excess. We further know that photons are emitted when the electron is attracted to the ion.
And we observe that the electron shell has almost no electric field towards the outside.
We know that static magnetic fields do not interact with static electric fields, but we observe the deflection of a forward moving electron in a magnetic field. And in the process, as with the attraction of the electron toward the nucleus of the atom, electromagnetic radiation is emitted. tldr
Field lines are used to illustrate how the field strength is distributed around a source. In calculations, however, a continuum is assumed and mathematically solved with gradients. However, are fields in the elementary scale really continuous or are they discontinuous, i.e. composed of quanta?
An indication that fields are quantized is given by the Feynman diagrams. In them, the interaction between the fields of electrons is explained by the exchange of virtual photons.

At the same time, however, it is pointed out that virtual photons, just like field lines, serve only for illustration and do not exist.
The terms field line and virtual photon introduced in physics shows our insufficient knowledge about the interaction of fields on elementary scale. Only the quantization of the fields can help here, i.e. the introduction of new elementary particles, whose sequence to each other can describe electric fields, magnetic fields and photons.
That it must be the goal to include photons in this description is obvious. First because of the above described photon emission of particles and - even more serious - second because of the structure of the electromagnetic radiation with its electric and its magnetic field component.
How do fields (electric, magnetic or gravitation) push any other particle of any object.
This is described by the fields. Your question about the more detailed description on a more elementary scale is not answered by physics yet.