Although we almost never saw any of them in Europe, there were later models in the Z80 family.
The first successors, the Z8000 (1985, 16-bit) and its later successor the Z80000 (1986, 32-bit) were not Z80-compatible. They did not do well.
Zilog did learn, though, and the contemporaneous Z800, which was Z80 compatible, was renamed the Z280 and relaunched in 1987. 16-bit, onboard cache, very complex instruction set, could handle 16MB RAM.
Hitachi did the HD64180 (1985), a faster Z80 with an onboard MMU that could handle 512 kB of RAM. This was licensed back to Zilog as the Z61480.
Then Zilog did the Z180, an enhancement of that, which could handle 1MB RAM & up to 33MHz.
That was enhanced into the Z380 (1994) -- 16/32-bit, 20MHz, but not derived from and incompatible with the Z280.
Then came the EZ80, at up to 50MHz. No MMU but 24-bit registers for 16MB of RAM.
Probably the most logical successor was the ASCII Corp R800 (1990), an extended 16-bit Z800-based design, mostly Z80 compatible but double-clocked on a ~8MHz bus for ~16MHz operation.
So, yes, lots of successor models -- but the problem is, too many, too much confusion, and no clear successors. Zilog, in other words, had the same failure as its licensees: it didn't trade on the advantages of its previous products. It did realise this and re-align itself, and it's still around today, but it did so too late.
The 68000 wasn't powerful enough to emulate previous-generation 8-bit processors. Possibly one reason why Acorn went its own way with the ARM, which was fast enough to do so -- the Acorn ARM machines came equipped with an emulator to run 6502 code. It emulated a 6502 "Tube" processor -- i.e. in an expansion box, with no I/O of its own. If your code was clean enough to run on that, you could run it on RISC OS out of the box.
Atari, Commodore, Sinclair and Acorn all abandoned their 8-bit heritage and did all-new, proprietary machines. Acorn even did its own CPU, giving it way more CPU power than its rivals, allowing emulation of the old machines -- not an option for the others, who bought in their CPUs.
Amstrad threw in the towel and switched to PC compatibles. A wise move, in the long view.
The only line that sort of transitioned was MSX.
MSX 1 machines (1983) were so-so, decent but unremarkable 8-bits.
MSX 2 (1985) were very nice 8-bitters indeed, with bank-switching for up to 4MB RAM, a primitive GPU for good graphics by Z80 standards. Floppy drives and 128 kB RAM were common as standard.
MSX 2+ (1988) were gorgeous. Some could handle ~6MHz, and the GPU has at least 128 kB VRAM, so they had serious video capabilities for 8-bit machines -- e.g. 19K colours.
MSX Turbo R (1990) were remarkable. Effectively a ~30MHz 16-bit CPU, 96 kB ROM, 256 kB RAM (some battery-backed), a GPU with its own 128 kB RAM, and stereo sound via multiple sound chips plus MIDI.
As a former Sinclair fan, I'd love to see what a Spectrum built using MSX Turbo R technology could do.
Two 6502 lines did transition, kinda sortof.
Apple did the Apple ][GS (1986), with a WD65C816 16-bit processor. Its speed was tragically throttled and the machine was killed off very young so as not to compete with the still-new Macintosh line.
Acorn's Communicator (1985) also had a 65C816, with a ported 16-bit version of Acorn's MOS operating system, BBC BASIC, the View wordprocessor, ViewSheet spreadsheet, Prestel terminal emulator and other components. Also a dead end.
The 65C816 was also available as an add-on for several models in the Commodore 64 family, and there was the GEOS GUI-based desktop to run on it, complete with various apps. Commodore itself never used the chip, though.