I've had 2 DC go through 13+. DS1 initially moved to senior school at 11+, but that turned out not to be a good fit so he went back to prep school, and took direct-entry 13+ exams in Y8. DS2 took 13+ pretests in Y6 and then CE in Y8.
We found the transition to senior school much smoother at 13+ than 11+. DC were more independent so travel to school, finding their way around a larger campus, organising their own extracurricular schedule etc were all much less of a challenge. We also found that both DC had outgrown their prep by the end of Y8 and were well and truly ready to move on, as were most of their friends. However, the 13+ process felt higher-stakes than 11+ because there wasn't the safety net of staying on at prep school if they didn't get a place at a school of choice.
13+ entrance processes vary from school to school. Some only do 13+ pretests, some do direct entry exams in Y7 or Y8, others do a mix of both. Some have their main entry at 11+ and then a smaller cohort join at 13+, others schools only start at 13+. We found that some schools advertised a 13+ intake on top of 11+, but in practice only admitted a tiny handful of students at 13+ and were not a realistic prospect unless your child was a scholarship candidate (and even then it was pretty much a lottery!). It's a good idea to quiz schools on entry numbers at 13+ before paying an application fee and putting your child through entrance exams.
Each entry process has its benefits and drawbacks. With direct entry via Y8 exams, you have more time to see how your DC develops and which school might be the best fit, but you don't have a senior school place until half-way through Y8 which can feel quite uncertain, especially if your DC is surrounded by classmates who've had conditional places via pretest since Y6. One plus is that CE becomes unnecessary (the prep school may still make them sit it, but it isn't crucial for the next stage).
Entry via Y6 pretest gives some security early on if your DC is offered a place at a school you like, but you can end up in limbo if offered more than one school or if you are on waiting lists. It can also be difficult to look ahead and be confident that the senior school you think will suit your 10/11-year-old will still be a good fit when they're a moody teenager! Common Entrance shouldn't be a big hurdle if your prep school goes to Y8 and is used to preparing pupils for CE, and there is often some leeway from senior schools because they know what a disaster it is for families to have a conditional place withdrawn so late in the process. DS2 had an unexpected blip in one CE subject but still had his conditional place confirmed.
Unless your child is very sociable, I'd recommend going for a school which either starts at Y9 or takes a large group at 13+ on top of 13+ entry - otherwise it can be hard to break into existing friendship groups in a group that's been together since Y7.