TL;DR : I'm not sure!

(In the following, I basically elide the difference between (weak) factorization systems and classes of morphisms defined by (weak) lifting properties, which is often harmless, e.g. if one works in a locally presentable category and throws in a small-generation assumption. Basically whereever I say "(weak) factorization system", feel free to substitute "dual classes of morphisms defined by (weak) lifting properties".)

I think there are relatively few interesting full-blown model categories where the fibrations or acyclic fibrations are closed under cobase-change (although upon reflection, the projective model structure on chain complexes of $R$-modules does have the property that the fibrations (=levelwise surjections) are closed under cobase change, for any ring $R$ -- so perhaps I'm mistaken!). I think it's fruitful to consider the weaker version of the question, asking for weak factorization systems whose right class is closed under cobase-change.

A strengthening of the dual situation has a name: a modality is defined to be an orthogonal factorization system whose left class is closed under base change. Many interesting factorization systems are modalities, and in fact some of them are very naturally thought about in terms of their duals. For instance, formally etale maps of (affine, say) schemes are basically the left half of a modality; the dual statement is that formally etale maps of rings are the right half of a factorization system which is closed under cobase-change. I have the sense that many of the factorization systems appearing in algebraic geometry follow a similar pattern.

I find this phenomenon mysterious and interesting. For another example, consider the category $Set$, which admits a factorization system (epi,mono), but also a weak factorization system (mono, epi). This tells us that from the perspective of the (epi,mono) factorization system, the epis are unexpectedly closed under base change / the monos are unexpectedly closed under cobase-change, and vice versa from the perspective of the (mono,epi) weak factorization system. I asked a question about this "interlocking factorization system" phenomenon here but didn't arrive at anything conclusive.

In the context of that question, I did spend some time thinking about the following version of your question: given a class of morphisms $\mathcal M$ defined by a weak right-lifting property with respect to a set of morphisms $I$, what conditions on $I$ will ensure that $\mathcal M$ is closed under cobase-change / coproducts / transfinite composition, etc. I arrived at only a few fragmentary observations, such as:

• If we work in a stable $\infty$-category, and if the codomains of the morphisms of $I$ are all the zero object $0$, then $\mathcal M$ is closed under cobase-change.

TL;DR : I'm not sure!

(In the following, I basically elide the difference between (weak) factorization systems and classes of morphisms defined by (weak) lifting properties, which is often harmless, e.g. if one works in a locally presentable category and throws in a small-generation assumption. Basically wherever I say "(weak) factorization system", feel free to substitute "dual classes of morphisms defined by (weak) lifting properties".)

I think there are relatively few interesting full-blown model categories where the fibrations or acyclic fibrations are closed under cobase-change (although upon reflection, the projective model structure on chain complexes of $R$-modules does have the property that the fibrations (=levelwise surjections) are closed under cobase change, for any ring $R$ -- so perhaps I'm mistaken!). I think it's fruitful to consider the weaker version of the question, asking for weak factorization systems whose right class is closed under cobase-change.

A strengthening of the dual situation has a name: a modality is defined to be an orthogonal factorization system whose left class is closed under base change. Many interesting factorization systems are modalities, and in fact some of them are very naturally thought about in terms of their duals. For instance, formally etale maps of (affine, say) schemes are basically the left half of a modality; the dual statement is that formally etale maps of rings are the right half of a factorization system which is closed under cobase-change. I have the sense that many of the factorization systems appearing in algebraic geometry follow a similar pattern.

I find this phenomenon mysterious and interesting. For another example, consider the category $Set$, which admits a factorization system (epi,mono), but also a weak factorization system (mono, epi). This tells us that from the perspective of the (epi,mono) factorization system, the epis are unexpectedly closed under base change / the monos are unexpectedly closed under cobase-change, and vice versa from the perspective of the (mono,epi) weak factorization system. I asked a question about this "interlocking factorization system" phenomenon here but didn't arrive at anything conclusive.

In the context of that question, I did spend some time thinking about the following version of your question: given a class of morphisms $\mathcal M$ defined by a weak right-lifting property with respect to a set of morphisms $I$, what conditions on $I$ will ensure that $\mathcal M$ is closed under cobase-change / coproducts / transfinite composition, etc. I arrived at only a few fragmentary observations, such as:

• If we work in a stable $\infty$-category, and if the codomains of the morphisms of $I$ are all the zero object $0$, then $\mathcal M$ is closed under cobase-change.