Let $\mathcal{C}$ be a category internal to (some convenient model for) topological spaces (which I will denote by $\mathsf{Top}$). In the question Greg Arone asks:

What is the correct notion of a topological functor $\mathcal{C} \to \mathsf{Top}$?

In particular, Greg wanted to know about the homotopy theory of such functors. My questions can be seen as somewhat of a follow-up question. Let $\mathcal{M}$ be a topologically enriched category.

What is the correct notion of a topological functor $\mathcal{C} \to \mathcal{M}$?

Particular examples I am interested in include (some convenient model for) based topological spaces and spectra.

I am also interested in understanding the homotopy theory of such functors.

If, in addition, $\mathcal{M}$ is a topological model category, what do we know about the homotopy theory of topological functors $\mathcal{C} \to \mathcal{M}$?

I am aware of one definition of such functors in the literature. In "Derivatives of embedding functors I: the stable case" Arone makes the following definition (Definition 3.1), which I will state only for based spaces, but Arone also defines for spectra.

A functor from a small topological category $\mathcal{C}$ to the category of based spaces consists of the following data:

1. A ex-space $F$ over the space $ob(\mathcal{C})$ of objects of $\mathcal{C}$, the fiber over $c \in \mathcal{C}$ is what we would usually call $F(c)$.
2. A fiberwise map of objects over the space $mor(\mathcal{C})$ of morphisms of $\mathcal{C}$ $$ \alpha: s^* (F) \to t^*(F), $$ where $s^*(F)$ and $t^*(F)$ are the pullbacks of $F$ from $ob(\mathcal{C})$ to $mor(\mathcal{C})$ along the source and the larget maps respectively.

This data is subject to unicity and composition law conditions, which I leave to the reference. The idea is that when $\mathcal{C}$ is discrete, this precisely recovers the standard definitions.

I'm hopeful that there is some 'slicker' way to define such functors analogous to that of a $\mathsf{Top}$-internal diagram as in Emily's answer to Greg's original question.