Your file size might be exceeding nginx’s client_max_body_size, @cduguet.
You can try increasing it in your app/conf/nginx/nginx.conf.hbs file. Change this line:
client_max_body_size 1000M;
To this:
client_max_body_size 4000M;
Then stop and start your site in Local.
If that does not work you could try importing your backup using WP-CLI from Local’s site shell using the WP-CLI commands for All-in-one migration, which should bypass any server limits (but not PHP’s limits).