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Although this question might be formulated in higher generality, let me try to be concrete:

Let $(\mathbf{Top},\times,*)$ be the monoidal category of compactly generated weak Hausdorff spaces; and let $\mathbf{C}$ be a small and $\mathbf{Top}$-enriched category (in the easiest case I have in mind: a topological group).

Now let $A\colon \mathbf{C}^{\mathrm{op}}\to \mathbf{Top}$, $A'\colon\mathbf{C}^{\mathrm{op}}\to\mathbf{Top}$, and $B\colon\mathbf{C}\to \mathbf{Top}$ be three enriched functors, and let $\vartheta\colon A\Rightarrow A'$ be a natural transformation. I am looking at the induced map between the enriched coends $$\int^{c\in\mathbf{C}}\vartheta_c\times\mathrm{id}_{Bc}\colon \int^{c\in\mathbf{C}}Ac\times Bc\to \int^{c\in\mathbf{C}}A'c\times Bc.$$ Assume that $\vartheta$ is an equivalence, in the sense that each $\vartheta_c\colon Ac\to A'c$ is a weak equivalence. Under which extra conditions can I conclude that also the induced map $\int^c\vartheta_c\times\mathrm{id}_{Bc}$ is a weak equivalence?

Some conditions I could imagine and which I would be happy with could be:

  • $A$ and $A'$ are free in the sense that for each $c\in\mathbf{C}$, the group $\mathrm{Aut}_{\mathbf{C}}(c)$ acts freely on $Ac$ and $A'c$,
  • for each morphism $f\colon c\to c'$ in $\mathbf{C}$, the map $Bf\colon Bc\to Bc'$ is an (equivariant?) cofibration.

Let me finish the question with some examples I had in mind:

  • If $\mathbf{C}=G$ is just a topological group, and we write $A$, $A'$ and $B$ for the corresponding $G$-spaces, then we just look for the induced map $A\times_G B\to A'\times_G B$, and here if is e.g. suffient that $G$ acts freely on both $A$ and $A'$.

  • If $\mathbf{C}$ is the semisimplex category $\mathbf{\Delta}^+$ of finite ordered sets $[n]=\{0<\dotsb<n\}$ and injective monotone maps, and if $B([n])=\Delta^n$, then it is well-known that the coend construction, which agrees with the ‘fat’ geometric realisation of $A$ resp. $A'$, turns levelwise weak equivalences into weak equivalences, see Ebert–Randal-Williams, Thm. 2.2

  • If $\mathbf{C}$ is the simplex category $\mathbf{\Delta}$, then we additionally need that both $A$ and $A'$ are proper.

  • If $\mathbf{Inj}$ is the category of non-negative numbers $r\ge 0$ together with injective maps $\{1,\dotsc,r\}\to \{1,\dotsc,r'\}$ and $\mathscr{O}=(\mathscr{O}(r))_{r\ge 0}$ is an operad with a prefered nullary in $\mathscr{O}(0)$, then $\mathscr{O}$ gives rise to a functor $\mathscr{O}\colon\mathbf{Inj}^{\mathrm{op}}\to \mathbf{Top}$. On the other hand, each based space $X$ gives rise to a functor $X\colon\mathbf{Inj}\to \mathbf{Top}, r\mapsto X^r$, and the free $\mathscr{O}$-algebra over $X$ is calculated as $\int^{r\in\mathbf{Inj}}\mathscr{O}(r)\times X^r$. Now if $\varphi\colon \mathscr{O}\to\mathscr{O}'$ is a morphism of operads, we get a morphism of free algebras $\int^r\mathscr{O}(r)\times X^r\to \int^r\mathscr{O}'(r)\times X^r$, which should be the unit of the base-change adjunction $\varphi_!\dashv \varphi^*$. Now if $\mathscr{O}$ and $\mathscr{O'}$ are $\Sigma$-cofibrant and each $\varphi_r$ is a weak equivalence, then the induced map on free algebras is a weak equivalence if $X$ is cofibrant, see in much higher generality Berger–Moerdijk, Prop. 5.7.

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One sufficient condition is that either $B$ or both $A$ and $A'$ are cofibrant in the projective model structure.

If $C$ is a Reedy category, possibly in the generalized sense defined by Berger and Moerdijk, and $A$, $A'$ and $B$ are cofibrant in the Reedy model structure, this is sufficient too.

The point is that under any of these assumptions the coend is equivalent to the homotopy coend, and therefore is homotopy invariant.

Im sure that this is well-known. I would think that a proof can be found in the book of Riehl on Categorical Homotopy Theory and/or Shulman's paper on Homotopy limits and colimits and enriched homotopy theory. But I don't have time now to try to find it.

As it happens I (co)wrote something about it. There is a discussion of enriched homotopy coends in Section 2 of this paper (see Lemma 2.9 and Remark 2.11). The case of generalized Reedy categories is Lemma 5.3 of this paper

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