Dump it or carry it?
Air, water, or land whatever be the medium, it is a well-known fact that there are restrictions on hazardous chemicals when it comes to traveling.
Talk about any solvent or paint base, there is some sort of risk involved with it. Therefore, it is either banned or is allowed in small quantities.
For standard allowance of such art materials and tips on their packaging walk by the dots :)
Paintbrushes: Once you start painting, you can never get an end of buying these. Apart from quantity and type, the quality of paintbrush matters as it is solely responsible for putting paints on canvas, blending them, etc. (apart from pallette knives... covered in the next point). It is important to keep paintbrushes for oil paints and acrylic paints separately. The reason behind it is that acrylic paints are water-based while oil paints are oil-based and there are different mechanisms to wash each of them out of brushes. Paintbrushes are allowed in your baggage while traveling (check-in baggage in flights) and are safe to keep anywhere under any weather and climatic conditions. All you need to care about is they should be washed properly and are free of any solvent whatsoever. Also, make sure they are wrapped properly especially the pointed side to avoid it poking through stuff.
Palette knives: Generally, palette knives come with round or soft edges. Although they play a role in sculpting and giving texture to the painting, they do not come in the category of sharp tools. They are allowed in check-in baggage in flights and are safe to travel without any hurdles. Again, keep them clean and free of any solvents.
Canvas, art surfaces, mixing palette, and easel: Its but obvious to carry all the canvas, boards whether they are painted or yet to be painted (Deets on how to pack them refer to Part-1 of this blog). The mixing palette can be carried if it's in workable condition else you don't need it to mess with other stuff so dump it. I always advise on using paper mixing palette easily available online because it's not only disposable but use and throw or frame-able :) Easels need a good layer of packing after folding and shrink sizing it to take the least space. Packing paper and bubble wrap with lots of taping are good.
Paints: Watercolors and acrylic paints are a no issue zone when it comes to restrictions except a lot of big bottles should be avoided. Oil paints have an oil base and may face strict restrictions when it comes to shipping or air travel because of the nature of paints and base. In unfavorable conditions, it's very easy for them to catch fire and lit up the shipment. So, if allowed, small tubes of oil paints can be carried. Remember NO LOOSE PAINTS.
Solvents: It is highly recommended to avoid carrying any solvent while traveling and dump them following all safety instructions before travel (Reference: Our blog post on how to handle solvents). Some of these include turpentine oil, alkyd mediums, liquin , varnish, etc. DO NOT CARRY ANY RUGS OR TOWEL DAMP WITH SOLVENTS AS THEY ARE COMBUSTIBLE AND CATCH FIRE WHEN THERE IS NO SPACE FOR SOLVENT TO EVAPORATE.