Western Punk fashion has been around for a very long time, in fact, it emerged in the late 70s and was more prominent in the beginning of the 80s. This subculture greatly influenced punk fashion in Japan and some of the substyles that emerged in the said country. One of them is a subset under a popular Japanese street fashion called Lolita, which was influenced by Japanese and Western punk fashion. Punk Lolita carries the typical Lolita silhouette while mixing it with punk elements like plaids, studs and leather pieces. Though it is not necessary to follow the volume of the skirts, but the idea of the look is still there.
Though many style of punk Lolita follows the basic bell or A-line shape of a typical Lolita fashion, it can also be toned down into something more casual, like frilly skirts with tartan patterns. But if they're going to follow the usual silhouette, they would choose Lolita dresses that do not require blouses, they are usually black in color with dark color accents like blood red or deep purple, or something with punk related patterns such as plaids or skulls. And unlike other Lolita substyles which requires even and precise designs, frills and cuts, punk Lolita dresses, tops or skirts can have uneven looks, some of them may have ripping which are forcibly sewn together with ribbons or with huge bobby pins. The skirts can also be uneven in the hemlines or it can be short in length layered with another fabric which are longer, giving off an uneven style which is very punk.
Since the typical Lolita requires to put on some blouse underneath jumper skirts, punk Lolita is a tad different than the rest. They can wear full on dresses as long as they have punk related prints and patterns. It can be anything with leather belts, customized with studs and lots of black ribbons. Frills are still present but with punk elements which makes it very unique. Gothic Lolita is also on the dark side of this fashion but it is seen to be elegant, like classic Lolita, punk on the other hand is very casual because you can wear skirts over graphic punk tees as well if you don't feel like putting on a Lolita dress. A simple layering of a punk shirt, fluffy and frilly black or punk skirt and some fishnet stockings and you're good to go, so you can see also that this Lolita substyle can be quite inexpensive than the rest that requires a lot of elaborate pieces and details.
Motifs can also be different from Western punk since punk Lolita tend to be a more "girly" version of the style, some themes would include religious artifacts like crosses, the crowns, roses, skulls of course, stars and sometimes, even butterflies.
As for footwear, unlike a typical Lolita fashion which often wear Mary janes or Ballerina platforms as shoes, punk Lolita is a lot different. They wear shoes that are typically associated with Japanese punk fashion or just punk fashion in general. That would include black platform boots with lots of straps, studs or other punk elements like mini skulls and crosses as decorations, the length also varies, it could be short or up to the knee.
Since we're talking about punk, accessories would mostly include studded cuffs, leather chokers, studded belts and arm warmers. Multiple earrings is also a thing for Punk Lolita, which includes different kinds of ear pieces like gothic chain earrings or something with a cross design. Silver chain necklaces are also an option for this look, some with typical punk pendants.
Hairstyle would also be quite simple, most punk Lolita just put up their hair into twin pigtails then accessorizing it with some Lolita head dress, some bows adorned with punk elements or a mini top hat. They can just wear their hair without needing to dye it dramatically or wear a bright colored wig just like some Lolita substyles, it really embodies the "casual trend" that its trying to project.
Punk makeup can have a lot of similarities with gothic Lolita, they apply their makeup dark but mainly focused on the eyes to make them look anime eyes big and kawaii (cute). Depending on the color of their dress or clothes, they can also wear their makeup a lot lighter to make them look cute and innocent, it's not necessary for them to make their faces look "dark" and "too gothic" for this substyle.
Compared to the other subnets of Lolita fashion, punk Lolita is the least common of the bunch, but since it can be put together with an extensive knowledge about the key elements you need to remember, it can be a quite inexpensive and easy. If you don't have any right knowledge regarding this substyle, you might just confuse it with another known fashion statement in Japan called Kawaii punk since they wear the same type of pieces (the casual, I mean), with shirts, skirts, some knee high socks and punk boots, but the difference between them are obviously the motifs because kawaii punk uses themes like cute cartoon characters and pastel colors mixed with the classic punk color which is black.