How does DNA damage cause ageing in yeast?
In eukaryotic cells there is no difference between a mother and a daughter cell - the later is an exact copy of the mother cell. This is true for yeasts as well for example for human cells. The only thing that happens over time is that the telomeres at the end of the chromosomes get shorter (unless the cell has an active telomerase which most cells doesn't) and it reaches the Hayflick limit where every further division will lead to a shortage of coding chromosmal regions and thus cause problems. If cells do not acquire any other chromosomal damage and have enough ressources, this would be the limit for their lifetime.

Of course there are DNA damages occuring over time. Most of them are either repaired by the highly efficient DNA damage repair system of eukaryotic cells, which consists of different checkpoints during the cell cycle. These checkpoints are located at transition between the different  phases of the cell cycle. If a DNA damage is identified, the cell cycle is stopped to enable a DNA damage repair by special proteins and to prevent that damaged cells undergo mitosis (here the G2-M checkpoint is important). This works schematically like in this figure (from the Wikipedia article on DNA repair):



If the DNA damage cannot be resolved, cells will either go into senescence (basically go into a permanent dormant state) or undergo apoptosis. If there are damages which are (for whatever reasons) are either not spotted or occur during the repair of DNA double strand breaks by non-homologous end joining, these are then replicated and inherited by further generations. Unless these errors occur during the replication process mother and daughter cells are genetically identical.

So eukaryotic cells acquire DNA damage over time (one of the reasons why late pregnancies pose a greater risk than early), but we are different from yeast in one point: Yeast has an active telomerase, which is not active in most of our cells (mainly except in stem cells). If the cells collect too much chromosmal damages than they will die.

The references listed below go deeper into the topic and are interesting to read.

References:


Negative regulation of yeast telomerase activity through an
interaction with an upstream region of the DNA primer
Cell cycle control in the kidney.
The evolution of diverse biological responses to DNA damage:
insights from yeast and p53
DNA damage and decisions: CtIP coordinates DNA repair and cell cycle
checkpoints
Cellular Responses to DNA Damage: One Signal, Multiple Choices