How to switch SMD part for through-hole part in EasyEDA?
After googling as recommended in Tim Wescott's comment I find it was easier than I had thought.
The basic process was

Switch to schematic tab
Click on the component whose footprint you want to change.
On the right hand side is now a side-bar with &quot;Component Attributes&quot; and &quot;Custom Attributes&quot;. Under &quot;Custom Attributes&quot; is &quot;Footprint&quot; showing SOIC-8... You can click in that and type DIP-8 and press enter to get a pop-up window with a list of matching footprints.
Clicking on each match gives a picture of each footprint. It also shows how the schematic symbol's pins now relate to the new footprint pads. This seems to be done automatically, perhaps by matching pin numbers.
You can select a suitable footprint.
Save the file
Switch to the PCB view
From the Design menu select Import changes
You then have to reconnect the tracks by dragging the control points on them.

I didn't have to do anything in the schematic to reconnect nets etc. Reconnecting the tracks in the PCB view was less work than I thought.
As Mat commented, the problem was in me selecting the wrong part number in the first place, The default library in EasyEDA only had SOIC and QFP parts for the ATTiny85. To do this right next time, I will find out how to add new parts to the library.