In all 3 of the non-Latin-script languages I have learned to relative degrees of fluency, handwritten and printed forms differed significantly. Of course, this should not come as a surprise. Often, learning handwritten forms required a significant amount of practice. Of course, being a language geek, I took pleasure in it. But as for the study of handwriting itself, to which branch of linguistics does it belong? I'd like to read some scholarly stuff on this. Maybe there is even a methodology for sample collecting.
I think Graphetics is the closest thing you'll get in Linguistics that among its studies subjects has the handwritten language.
As stated in the article: "Graphetics is to the study of writing as phonetics is to the study of spoken language. As such, it can be divided into two areas, visual graphetics and mechanical graphetics, which are analogous to auditory and articulatory phonetics, respectively. Both printed and handwritten language can be the subject of graphetic study."
Emphasis and bold are mine.
As mentioned above, anything that is related to the study of handwriting in historical documents is part of paleography. Paleography is a large discipline that covers all kinds of writing systems from widely different periods, and it includes the study of handwritten (as opposed to printed or epigraphical) writing. Handwritten forms of writing are often labelled as cursive or running writing, i.e. scripts typically used for everday purposes and/or for private correspondence. The letters of cursive writing tend to be "sloppier" than printed ones and they tend to be interconnected much more frequently.
For the scientific study of (past and present) writing systems in general there are the terms graphemics and graphetics. As in phonemics vs. phonetics, the former one is mainly concerned with the system of distinctive signs, while the second one is concerned with how those distinctive signs appear in actual scripts.
Graphemics and graphetics are not to be confused with graphology, the latter being a pseudo-scientific method of drawing conclusions about someone's psychology by analysing their personal handwriting.
There is to my knowledge no specific term for the scientific analysis of handwritten (as opposed to printed) writing. If you are looking for more information, your best bet is probably to search for cursive writing of whatever script you are interested in.
Despite the existence of a name for a subfield ('graphetics') and a reasonable expectation of study, the currently accepted faculty of linguistics as practiced in most universities these days does not consider handwriting as part of their domain; that is, linguists on the whole do not tend to study and discuss handwriting. Linguistics as currently practiced deals only with spoken language).
The faculties where handwriting is studied in general are cognitive psychology (with respect to how people read it) or clinical psychiatry (evidence of problems shown in handwriting phenomena (the respectable arm of graphology), computer vision (recognition of handwritten characters), or history/archaeology (deciphering of old scripts). Written language (where handwritten script is a special, but not specially considered, case, is studied in literature departments.
If you want to find out about handwriting academically, I suggest looking to foreign language teaching literature for the individual scripts concerned. For example, look in instruction material for Hebrew, in English, that describes how one is supposed to draw the script version of characters. As that might be too practical, more scholarly discussion of handwriting will probably be limited to a foreign language teacher's special monograph, that is, most likely -not- a linguist but a specialist in one particular language/writing system.
Graphemics/graphetics is more of a sibling field to linguistics than a child. Their parent would be semiotics: spoken language represents ideas, written language represents spoken language (usually). So maybe graphemics/graphetics is more of a niece or nephew since it's dependent on the existence of spoken language...
the study of handwriting itself, to which branch of linguistics does it belong?
This kind of hierarchical thinking is fundamentally flawed. The concept of university ought to some extent be self-descriptive.
On the risk of commiting a true etymological fallacy to the extent that it is fallacious if and only if the etymology is wrong, I would argue that the hiero- ("holy") in Hieroglyphics is a doublet of chiro- ("hand") because monumental, epigraphic hieroglyphs are disproportionately overrepresented in contrast to the Hieratic papyri. The hieroglyphs make this quite clear: 𓏛 mḏꜣt (Gardiner Y1 "papyrus roll") 𓊹𓌃 mdw-nṯr ("word of god") 𓌃𓂧𓅱𓀁 mdw (“word; speech; text; command; plea”) 𓏞 zẖꜣ (Gardiner Y3 "scribe's equipment") zẖꜣw-mdw-nṯr ("Egyptian hieroglyphic writing"), cp. Sumerian dub "tablet", dub-sar "writer", Greek diphthera, Mycenaean 𐀇𐁇𐀨, Latin letter, etc. but Luwian 𒄿𒅖𒊭𒊑𒅖 /īssaris/, Old Persian d-s-t /dasta/ "hand"; hiero- is uncertain but PIE *ser- has been suggested.
The cursive Hieratic hands make this much less clear, which is an actual problem of epic proportions.
This does, admittedly, not amount to defeating evidence. It is still a worthwhile exercise to try and pinpoint the origins of χ chi as in χειρ “hand” as opposed to other hand signs. Psychology is necessary prerequisite because those guys were cray-cray.
A related topic which other answers haven't mentioned is forensics. If you are researching whodunit and you cannot determine a suspect because they religiously cross all the t's and dot the i's, forensic linguistics may be able to help.
Graphetic is potentially a misnomer, if following the the etic / emic distinction, i.e. "Of or pertaining to analysis of a culture from a perspective situated outside all cultures" (en.WT: “etic”), that was modeled on the example of phonetic / phonemic.
NB: Meletis & Dürscheid (2022) For the difference between graphetics and graphematics, as pointed out by Seyr (2023, in: Hierpglyphs 1)
Meletis, D. & Dürscheid, Ch. 2022. Writing systems and their use: an overview of grapholinguistics, Trends in Linguistics, Studies and Monographies 369. Berlin & Boston, De Gruyter & Mouton.