Bringing cremains with you on a commercial flight, train, or bus ride is possible, but it has to be planned carefully. Spilled or lost cremains are not what you want to deal with on your way home or to wherever you're scattering the ashes. The best way around these issues is to choose a travel urn, which can be a temporary urn that is made solely for the trip (they're commercially available, but they are no-frills and meant for traveling, not display), or a permanent urn with travel features built in.
If you're flying, the urn has to allow TSA officers to see what's in the box when it passes through the x-ray machine. If the picture is opaque, the urn would have to either be checked, if the airline allowed that, or left behind if the airline did not allow it -- and it's true, some do not allow cremains in checked baggage. Save yourself the drama and get an urn that will give TSA officers a clear x-ray picture. They will not open the urn.
The lid on the urn must close securely. You should not take an ornamental urn with a loose lid on any sort of trip -- even if you're only moving house a few miles away, transfer the ashes into a travel urn that you can seal. If you accidentally tip over the urn, then the top won't fall off.
The base of the urn should be very stable. A rectangular box is a good choice; ornamental vase-style urns can be unstable and fall over if everything in your luggage shifts around. While you could pack your bag so snugly that the urn is wedged in place, the jostling that your luggage can get while traveling could be enough to compress and deform the bag, ruining your secure packing job.
The urn needs to be labeled. If you place the urn in your luggage plain, someone trying to rifle through your bag could think it was a jewelry box and take it. If the box is clearly labeled as cremains, someone might think twice before trying to run off with it.
The crematorium or funeral home that handled the services should have a selection of urn styles including those meant for travel. Take a look and see whether you want a separate travel urn or a permanent one with travel-friendly features. For moe information, contact a company like Holmes Funeral Home.