Some notations:

1. $\mathbf{T}$ is the category of weak Hausdorf $k$-spaces.
2. $\mathbf{K}$ is the category of $k$-spaces.

Fact: The inclusion functor $\mathbf{T} \subset \mathbf{K}$ is a right adjoint. It does not preserve colimits in general. However, it preserves pushouts of closed inclusions and transfinite compositions of closed inclusions (in fact, the closedness hypothesis can be removed for transfinite compositions).

Observation: The common point between the two colimits preserved by the inclusion functor $\mathbf{T} \subset \mathbf{K}$ is that it exists a degree function (raise the degree where the closed inclusions are) so that the base small category is a Reedy category which has fibrant constants (which implies that in both case, the colimit functor is a left Quillen functor).

Does this situations have a generalization ?

The most naive generalization I can think of is: for any small diagram $D:I\to \mathbf{T}$ such that $I$ is a Reedy category which has fibrant constants, if for all $f\in \vec{I}$ (the direct subcategory), $D(f)$ is a closed inclusion, then the colimits of $D$ calculated in $\mathbf{T}$ and in $\mathbf{K}$ are equal.

Notations:

1. $\mathbf{T}$ is the category of weak Hausdorf $k$-spaces.
2. $\mathbf{K}$ is the category of $k$-spaces.

Fact: The inclusion functor $\mathbf{T} \subset \mathbf{K}$ is a right adjoint. It does not preserve colimits in general. However, it preserves pushouts of closed inclusions and transfinite compositions of closed inclusions (in fact, the closedness hypothesis can be removed for transfinite compositions).

Observation: The common point between the two colimits preserved by the inclusion functor $\mathbf{T} \subset \mathbf{K}$ is that it exists a degree function (raise the degree where the closed inclusions are) so that the base small category is a Reedy category which has fibrant constants (which implies that in both case, the colimit functor is a left Quillen functor).

Does this situation have a generalization ?

The most naive generalization I can think of is: for any small diagram $D:I\to \mathbf{T}$ such that $I$ is a Reedy category which has fibrant constants, if for all $f\in \vec{I}$ (the direct subcategory), $D(f)$ is a closed inclusion, then the colimits of $D$ calculated in $\mathbf{T}$ and in $\mathbf{K}$ are equal.