Some links to research on cursive handwriting:
What Learning Cursive Does to Your Brain
Cursive Benefits Go Beyond Writing
Brain Research and Cursive Writing
Should Schools Still Teach Cursive?
Study: Learning Cursive in First Grade Helps Students
Edited to add a new one:
Biological and Psychological Benefits of Learning Cursive
Each one of these studies focuses on learning cursive in approximately 2nd-4th grade (ages 7-9). The naysayers to cursive typically are the ones not realizing that the sensitive period for languages is 0-6 years of age, thus we should be giving the children the keys (muscle memory of the letter shapes, sounds of the letters) before age 6, at which time they receive all they need to read/write in cursive AND in print (because print just "happens" with children who start with cursive).
Then, we can spend those elementary years working on composition skills and pursuing SO many other interests pertinent to the elementary age - the second plane of development.
Why Cursive First? (fantastic bullet-pointed list)
Should We Still Teach Handwriting in the Montessori Classroom?
(read the comments on this one for sure!)
There are plenty more, very excellent, articles - and quite a few not so excellent ones (both Montessori and non-Montessori). But these ones hit the main ideas.