Treat it like any liquid.
You don't need much.
Just put some in a 100 ml plastic bottle if carrying on.
Alternately, if checking your bags, put your starter in a larger plastic container with a screw cap...like a Nalgene bottle, for example.
You can build it back up once you get to your destination.
cooking.stackexchange.com/users/17143/moscafj
If you don't want to transport liquid (= active) sourdough because you don't want to risk it to spill out of your container, note that you can either mix some of it with more flour than usual until you get a dryish crumbly mass that can be stored in a plastic box or bag or spread a thin layer on parchment and dry it.
Break up the "sheet" and store it in a dry place.
Both forms should be safe to be transported in either your checked baggage or in your carry-on luggage.
Be prepared to answer questions from security either way.
But I suppose liquid sourdough might raise the same questions.
At your destination, rehydrate the crumbs, add them to your usual water / flour mix and continue cultivating it as before.
A few feedings should be enough to get the sourdough back into good shape.
cooking.stackexchange.com/users/28879/Stephie
Disguise it as cookie - the airport does not generally take your food items unless they are forbidden when entering or leaving the Country.
Mix a small amount (10g) of your starter with about 3-4 times the amount of unbleached or whole grain flour.
Once you can no longer incorporate the flour press it flat to form a cookie - you can even add raisins to the disguise if you want.
Place it in a plastic bag and store it in your travel bag with your other snacks.
If asked just tell them its a macrobiotic cookie, which is pretty close to the truth...
Once you arrive at your destination remove any raisins that you might have added and break the cookie into pieces in a bowl - add water and flour as you would when refreshing your starter.
Let it ferment for 4 - 12 hours depending on how you normally use your starter - use or store as usual.
I usually make a counter starter occasionally from my refrigerated starter.
The recipe was 10g starter, 10g water, 20g of flour usually whole wheat and rye.
The end result will be something like this.
Sorry I don't have a cookie picture.
cooking.stackexchange.com/users/61732/millciti
TSA did NOT allow us to take a partially filled bottle even though it was a 3 ounce bottle.
If you check with TSA, write down who you talk to in addition to the date and time and what they recommend.
cooking.stackexchange.com/users/55329/Brendoodle
Using a wide mouth 1/2 pint jar...feed your starter in starter jar and then fill small 1/2 pint travel jar almost to top.
Leave uncovered for three days before travel.
The top of the 1/2 pint will dry over.
This "skin" will protect the nearly dessicate starter safely hiding under the skin.
In this way you will protect the lacto family of worker yeasts, which thrive in a moist environment, for the travel.
Once you arrive to your sestination, remove the protecting skin and revive your starter.
If you are a sourdough keeper you will recognize when they are ready to go to work on a fresh loaf of loveliness.
cooking.stackexchange.com/users/78592/Tyrus
We recently got a sourdough starter, and have loved trying out many different recipes with it.  In a few weeks we will by flying home (after spending the summer in Alaska), and we would really like to take the starter home with us.  What's the best way to get it safely through the airport?
Traveling with sourdough starter
cooking.stackexchange.com
<sourdough><sourdough-starter>
cooking.stackexchange.com/questions/83286
cooking.stackexchange.com/users/60494/David Oneill
You could try treating it as a liquid and putting it in a 100ml plastic bottle, or disguising it as a cookie since food is not normally confiscated. However, TSA may still not allow you to take it with you. A sourdough starter can be treated as a liquid or disguised as a solid when flying. However, some airlines do not allow it on board.
Treat your sourdough starter like a liquid: put in a suitable size bottle or pack with your cargo luggage. Disguise your starter as a cookie by adding a lot of flour. You may have problems when flying, dependingon the airline.