It is not true.
If a chain or rod is being pulled along it's length unconstrained it will accelerate.
There will be a force gradient along that length, greatest at the pulled end and zero at the other depending on the distribution of mass.
The force at any point will simply be proportional to the amount of mass behind that point. If the links are all the same, the force will be proportional to the number of links remaining behind the point in question.
The force is reduced between each successive link, due to the acceleration/reaction of the preceding link.
If you drew a free body diagram of any link, or set of links, the acceleration would be proportional to the net force, and inversely proportional to the mass of the link or links.
So if, say, you had a 10 lb force pulling a chain of 10 links of mass 1 slug each the acceleration would be 1 foot per second per second. The force on the last 1 slug link would be 1 lb and it would nicely accelerate at 1 foot per second per second along with the rest of the chain.
Or if, say, you had a 10 newton force pulling a chain of 10 links of mass 1 kilogram each the acceleration would be 1 meter per second per second. The force on the last 1 kilogram link would be 1 newton and it would nicely accelerate at 1 meter per second per second along with the rest of the chain.