When applying to a UK university nowadays, student normally fill out an online application where they will input their referees' email addresses/contact details.
The institution will then contact them automatically, usually via email, with some kind of form asking things like what percentile the students was in their class, give detail of their relationship with them, for how long etc. and sometimes a tick box 'would you recommend them for this course' question yes or no.
They would also be asked for qualitative data in the form of a letter of recommendation. The could be free text or upload for a PDF or word doc (in my exp. it's always both).
However, in my experience, the prompt for the recommendation will have specific questions related to the course/area of research - eg comment on their ability to research such and such, do they have potential to research such and such.
So, your referees would likely be asked to provide some kind of course dependent information anyway. Whether they copy and paste a reference from a previous letter, add a few lines to it or do a brand new one afresh is entirely up to them.
If you are applying for a PhD, I personally would expect you to have a fairly solid research topic and proposal in mind already. Therefore, your references are likely to all need to support you in more or less the same capacity, and of course, you are the same person, so have similar references across different applications would be expected.
The only difference could be if you are applying for a specific project that is very different to PhD applications you have already made. In which case, I would explain this to your referee and allow them to make up their mind as to whether they need to provide an extra bits to their ref doc for you.
Telling someone what to put in a reference is bad! But explaining if there is a key difference would be helpful - perhaps make it clear when you request it at the outset.
Finally, remember that your profs can be trusted to give an accurate and appropriate reference for you - they after all receiving references for other students, so they know what is expected!
Should my referees tailor their letters of recommendation to the specific programmes I'll be applying to, or is the convention usually for letters to be more destination-agnostic?
Leave it to your referee to decide, they will probably be asked context specific questions anyway, and the potential for variation is probably overstated.
Is it ok to ask a letter-writer to make the letter relevant to many degrees or asking for multiple separate letters is the usual practice?
If there is a genuine need, make it clear to the referee that there is a specific difference with or more applications that you are making. Don't tell them what to say, but highlight the areas of difference.