Does undergraduate school matter for PhD admissions (international student)?

I am sure some people will disagree with me, but this is what I can say as a person actually reading application documents and hosting international students (including PhDs).

First, your school name does matter, but to a lesser degree that one might expect. A "well-ranked school" on national level might be virtually unknown to people in another country. As a rule, top national schools plus top schools from international ranking lists ring the bell, but the rest is "large unknown", so it is very possible that school A in your list is not that familiar for the program's committee members as you may believe.

Second, we have to look at school names due to absense of better criteria. Consider this: tons of people send their applications to MSc/PhD programs. They all have reasonably high GPAs, they possibly had work experience, etc. What's the difference? OK, school name is one such factor that can make an application stand out.

However, there is a much better way. If you aim at PhD level, you must make your personal part of the application stand out. In practice it means that you should publish scientific papers, and the higher the rank of the respective journals/conferences, the better. Think how to strengthen your profile as a scientist, and you will greatly increase chances of admission anywhere in the world. Also don't think in terms of "sending students" to a program. It is your personal endeavor, and school's power in pushing you as a candidate for another university PhD program is usually limited.

(Of course, here I have to note that it is probable that a better school can be more effective in supporting your research activities, but this is another issue).