How long would you expect a designed plasmid to remain in a population of E. coli after transformation?
There isn't a simple answer.
There are a lot of factors which can influence how stable a plasmid will be. I'll describe what the main factors are, but to work out exactly how stable your plasmid is, unfortunately you'll just have to test it. I've given links to some studies which do this at the end.
Copy number
Copy number refers to how many copies of your plasmid are present in each cell. This can range from 1-10 for low copy number plasmids, to 300-500 for high copy number plasmids. Generally, low copy number plasmids can be lost easier than high copy number plasmids as there is a higher chance that the plasmid won't be present in one of cells after division. (https://www.nature.com/articles/s41598-017-05219-x)
Plasmid Size
Very large plasmids (in the mega base range) tend to be less stable than smaller plasmids as organisms, especially E. coli, can struggle to replicate them. This means that over time, less of the plasmid will be present in your system. (https://pubs.acs.org/doi/10.1021/acssynbio.0c00008)
Selection marker
Typically, synthetic plasmids will have a selection marker encoded which is used to provide selective pressure for keeping the plasmid. The exact marker used can affect the stability of the plasmid, as some are more effective than others. This paper has a good comparison of two different markers and measures the stability of plasmids for each: https://bmcbiotechnol.biomedcentral.com/articles/10.1186/1472-6750-8-61.
Burden
Plasmids normally contain an insert which encodes your system. Some inserts will have relatively low burden on the cell (for example low expression of a small protein). Others may involve high expression of many proteins, or proteins which are toxic to the cell. Mutations are much more likely in high burden plasmids compared to low burden plasmids. This is because cells containing plasmids with mutations which reduce/stop expression will have a much higher fitness advantage over those cells with unmutated plasmids. Other time, these mutated plasmids will become dominant and eventually you will lose your original plasmid. There's some good reading about cell burden here (https://2019.igem.org/Team:Austin_UTexas/Measurement) and here (https://pubmed.ncbi.nlm.nih.gov/25849635/).
This is anecdotal, but I've managed to run 24 hour experiments with plasmids about 5kb in size containing a low burden insert and without using any selection marker without seeing plasmid loss. I've also had trouble even cloning plasmids of about the same size but with antibiotic selection and a high burden insert without any mutations occurring. So it really does depend on what your plasmid is and what it's expressing.
Measuring plasmid stability
Here are some examples for how you can measure plasmid stability:

https://link.springer.com/protocol/10.1007/978-1-4939-9877-7_16
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3966108/ - This paper describes plasmid stability mathematically
https://academic.oup.com/femsle/article/232/1/45/490969