The bug
When teleporting a chest or hopper minecart between dimensions, the minecart drops its contents at its source point. It arrives at the destination entirely empty. This does not happen when the minecarts moves through a portal manually. Scoreboard scores are also lost, though the minecart's custom name and other data are maintained.
How to reproduce
Get a minecart with some items inside:
/summon chest_minecart ~ ~ ~ {Items:[{id:"diamond",Count:1b}]}
(optional) Give it a score:
/scoreboard objectives add test dummy /scoreboard players set @e[type=chest_minecart] test 5 /scoreboard objectives setdisplay sidebar test
Teleport it to another dimension
/execute in the_nether run tp @e[type=chest_minecart] 0 64 0
→ ❌ The item is left behind and the score is lost
Related issues
Comments

Items appear to make it over fine now but scores are still lost
Closing as fixed, the score issue is now tracked in MC-161592
Confirmed for 1.13.1.