Approaches to Model Building I often see references to 'model building' in the particle physics literature, presumably to refer to creating new QFTs which go beyond the Standard Model.
How exactly does this process of model building begin?  Does one simply write down a Lagrangian which has the desired properties and then alter it with trial and error, or if not, how does one arrive at the Lagrangian?  If one assumes the Standard Model gauge group is embedded in a larger gauge group like $SU(5)$ or makes some extension to the gauge group, how does that change all the terms going into the definition of the Lagrangian and the covariant derivative?
 A: I am writing this in an answer because comments tend to be deleted,  it is  a sort of answer from an experimental physics physicist.
One does not start from mathematical impulses to generate models. In physics the process is data driven, there exist data that the standard model cannot fit. This starts a search for a mathematical model that could do so. It is not always a different Lagrangian. For example neutrino oscillations were introduced to explain the neutrinos from the sun data, using the standard model Lagrangian.
A number of Lagrangians have already been explored in trying to fit data and predict more, so a new model builder that tries to fit  a new high mass resonance in the LHC data can choose among existing solutions that could fit it.
But it  is not just new QFTs that are explored but also new formats. These have to be able to embed the standard model lagrangian as it is a resume of most of the data up to now. Example : string theories.
Also there are creative proposals to fit strong interactions, like the amplituhedron.
A: From Anna’s answer:

One does not start from mathematical impulses to generate models. In physics the process is data driven, there exist data that the standard model cannot fit. This starts a search for a mathematical model that could do so. It is not always a different Lagrangian.

Sometimes the search for a mathematical model is the second step. The first step is the pure imagination of how something should work. Maxwells work is a good examples for that. Before developing the math he introduced a mechanical model of how electromagnetism works:

From this model to the math it was a long way.
Why I'm writing this. Because we use virtual photons to describe field interactions (which are never real or exist, as so often written on PSE). The quantization of these fields, let's say along field lines, is never envisaged at any time. Research into this will explain the understanding of subatomic processes - the stopping of the approach of electrons near the nucleus as opposed to the annihilation of a particle and its antiparticle.
Why I'm writing this. Because I'm like a musician who can't write notes. I'm not capable of writing high-level equations. There is a theory [Are photons composite particles] (https://www.academia.edu/11805855/Are_photons_composed_particles) only. This theory is based on a very dry written treatise on how field lines could be imagined as real with an inner structure, Complex one-dimensional structures of space.
The results are surprising and the predictions fit the Standard Model. The new point is the exchange of virtual photons by real interactions for electric fields, magnetic fields and the consistency of photons.
