The first single-bed machines to produce what is now referred to as knit-weaving were the Japanese model ones from the early to mid-1960s.
The Knitmaster 302 had a separate brush unit operated manually to guide the yarn above and below stitches.
On the Brother 585 and 588 models the weaving brushes were built-in and were pressed into place.
Early patterns used aran-weight yarns as weft on 12-stitch machines.
From 1971 onward as 24 stitch machines came on the market, the wider repeat allowed for the use of thinner yarns in the technique.
Later manuals including those for some electronic models describe this achingly slow method when using a single-color weaving yarn fed through the yarn mast 
The subsequent technique shift advice was made for using an intarsia-like method for more colors: 
The method is so much faster than when using the mast and knit carriage it makes little sense to use the former.
The technique begins with the carriage on your preferred side, from the right or left, the yarn is generally laid in starting on the carriage side, with the long end to cone or ball away from it. 
The wraps are needed in each direction: 
For work on the machine, swatches, punchcards, tools, and more information on types of weaves, ie laying in the yarn with vertical wrappings 
Another illustration of the movements in the laying in of the yarn on specific needles. 
For more info on horizontal weaving, see Knit Weaving 1.
The resulting knits often lose stretch and drape widthwise, a consideration when making garments. As an alternative, one may knit pieces sideways.
Some ways to push limits, and new swatches:
Knitweaving encompasses an extensive range of fabrics. At times the type of machine available places constraints on materials.
One way to use chunky and bulky yarns on standard machines when they do not work with the ground knit on every needle, an option is to use every other needle patterning.
A visual comparison of yarn thicknesses for beginning experiments 
Here a commercial twine is used, programming a 4X4 repeat, 
and laying the weaving yarn over the preselected needles when the long end of the yarn is away from the knit carriage, thus adding extra all-knit rows between woven ones. 
This yarn has been in my stash for decades unused, it is 92% wool, and 8% polyamide. It was e-wrapped in alternate directions on every row, with care to maintain the thrumbs below sinker plate levels during carriage passes. 
An option for electronic models: open a PNG suitable for knit weaving, as this 24X24 one, double the PNG in width before downloading it, or use the double-wide button after the fact, cast on every other needle, and continue in the technique. Planning the repeat on the needle bed would render better edges than in my swatch, if there are multiple non-selected needles on either side, bring the last needle on that side out to E before laying the yarn for the next row of pattern. 
The floats created through knit weaving can share similarities with those from fair isle or slip stitch.
Slip stitch textures compress the knit lengthwise, and narrow it, while knit woven pieces lie flat, and provide far more predictable gauges.
This sample explores various float widths, beginning with a 24X24 pixel PNG.
When sampling on an electronic, one can plan for repeat widths equal in number to the needles in work or greater based on preference or a requirement by some software.
In designing for planned fabric widths, borders for better side edges than those resulting only from engaging end needle selection may be isolated or added with a preferred pattern fill.
The 24-stitch repeat was drawn in repeat X4 
and cropped to 50X48, retaining a 2-stitch vertical border on each side and shared repeated in height X2, a final 50X48 
Samples of the three yarn thicknesses used in the test swatch are shown at the bottom of the image on the right
More links for browsing
A return to loopy knits
Machine knit fringes 4, long loop patterning
Long loops: a bit on method
Knit weaving 2: swatches, experiments
Knit weaving 1
Tuck stitch meets knit-weaving
Lace meets weaving on Brother Machines 2
Lace meets weaving on Brother Machines
2024: lately I have been obsessed with designing seasonal repeats and expanded my initial library of them to include more “star’ designs in Seasonal knits inspired by published repeats 1
This design, 36X56, was knit first using a worsted-weight acrylic as the weaving yarn and a 20/2 wool for the background, then replacing the worsted with a 2/8 wool.
The design has long floats that tend to droop in front of the small stitches in the red ground as the knit is removed from the machine and rests.
The white acrylic was hard to knit, resulting in patterning errors because of its quality, noticeable on both sides.
In a final piece, borders could be added to make side edges cleaner than in this first swatch. 
For use in a scarf or table runner, the repeat was altered slightly, and vertical side borders were added to 82X56
Visualizing the repeat multiplied in height 
The yarns used in the swatch were a 2/8 wool and a softly spun worsted, challenging to work with even at the loosest tension.
The worsted wanted to split and knit with the ground yarn while forming the floats in front of those same needles, occasionally mounting a few gatepegs or jamming.
The mispatterning in the lower part of the swatch improved as the numbers were slowly adjusted to more than 10 on the knit carriage dial.
This time of year static can also be a problem, the humidifier got turned on. Both steps led to better results.
As with any technique, good yarn choices, and knitting machine gauge considerations make for successful projects.
Working with needles out of work, NOOW
In this first sample, after a crochet cast on, every 3d stitch based on pattern preselection is dropped, and the corresponding needle (grey columns) is pushed out of work (OOW), back to the A position. Ladders will form in those locations on the needle bed.
The yellow cells mark the stitch columns left undisturbed, while the pink and blue mark the stitch/needle locations involved in the cable crossings. The latter are consistently made in the same direction.
After each cable is made, restore needle preselection to maintain correct patterning. 
The swatch may be knit monochromatic or as here, using contrasting colors