Brushes are damaged in transit more often than at the desk, because a point that survives a session will not survive being pressed against the inside of a bag. The four common answers are a travel brush with a retracting or reversible handle, a brush roll, a rigid brush case and a simple paint brush box, and they solve slightly different versions of the problem.
The travel brush, and what it trades
A travel brush protects its own point by retracting the head into the handle or by capping it, which means it needs nothing else around it. That makes it the tidiest solution for someone carrying one or two brushes to a club night or a painting session away from home. The trade is that the handle is usually shorter or differently balanced than a desk brush, and travel brushes are made in a narrower range of shapes and sizes. As a second set they work well; as your only brushes they are limiting.
Rolls against rigid cases
A brush roll holds brushes in fabric loops and rolls up, so each brush is separated and nothing presses on the tips. It is light, cheap and packs into a bag, and it offers no protection at all against something heavy landing on it. A rigid case does protect against crushing and is bulkier and more expensive. Painters who travel by car tend to prefer cases; painters carrying a bag on a train tend to prefer rolls, and both are right about their own circumstances.
The box that is not really storage
A simple paint brush box, or the tube a brush arrived in, will keep a brush from being crushed and is perfectly adequate for occasional transport. The mistake is treating it as permanent storage: sealing a brush that is not thoroughly dry into any closed container traps moisture against the ferrule, which loosens the glue and eventually sheds hairs. Whatever the brushes travel in, they should be dry before they go in it and should come out and stand tip-up when they get home.
Questions people ask about travel brush
How do I carry brushes without ruining them?
Anything that stops something pressing on the tips: a roll, a rigid case, or a travel brush that caps its own head. The tips must not touch anything solid.
Is a brush roll or a case better?
A roll is lighter and packs smaller; a case protects against crushing. Which is better depends on whether your bag gets sat on.
Can I store brushes in a travel case permanently?
Better not to. A closed case traps moisture against the ferrule if the brush is not fully dry, which loosens it over time. Store them upright in open air.