In September 2019, a large batch of errata was released by Donald X., the game's creator, to the so-called shapeshifter cards, including Inheritance. Also some other cards got errata to work more "as intended" with the new wordings of the shapeshifter cards. You can find the whole set of errata and additional rules changes here (original post and an update to prevent infinite loops)http://forum.dominionstrategy.com/index.php?topic=19893.0
I'll just copy the current text of Inheritance and Death Cart out if it (can be found in the online game as well):
Inheritance: Event, $7
Once per game: Set aside a non-Command Action card from the Supply costing up to $4. Move your Estate token to it. (During your turns, Estates are also Actions with "Play the card with your Estate token, leaving it there.")
Death Cart: Action - Looter, $4
You may trash this or an Action card from your hand, for +$5.
When you gain this, gain 2 Ruins.
As you can see, Inheritance no longer has the Estates "become" the inherited card, it makes (on your turn) Estates just Actions that play
the set aside card.
So in your case, an Estate plays the set aside Death Cart. In other words, if you want to trash the "this" mentioned in Death Carts on-play ability, you would have to trash the set aside Death Cart, not the Estate that played it.
This however is not possible, because the Death Cart expects itself to be in play, which it isn't. So your only chance to get $5 would be to trash an "Action card from your hand". This can be an Estate (the Inheritance effect made an Action card), or any other Action cards, but it can't be the one in play.
So some nice combos from the "old days" of Inheritance no longer work, unfortunately, and this is one. OTOH, gaining an Estate when you inherited Death Cart doesn't mean you get Ruins any more, which is generally a good thing but of course they were often good Death Cart fodder.