The accepted answer was correct (when it was written)

There was a subsequent discussion in github with the maintainer of the postgres docker image about supporting a mechanism similar to the mysql /always-init.d/ etc.

The link to that discussion: https://github.com/docker-library/postgres/pull/496

The solution with docker is to provide a custom entry-point, there is a bit of a bug on version shared in issue 496 so I'm posting a more updated version here in hopes others will find it useful:

#!/usr/bin/env bash
## copied from: https://github.com/docker-library/postgres/pull/496#issue-358838955
set -Eeo pipefail

echo " custom-entry-point"
# Example using the functions of the postgres entrypoint to customize startup to always run files in /always-initdb.d/

source "$(which docker-entrypoint.sh)"

docker_setup_env
docker_create_db_directories
# assumption: we are already running as the owner of PGDATA

# This is needed if the container is started as `root`
#if [ "$1" = 'postgres' ] && [ "$(id -u)" = '0' ]; then
if [ "$(id -u)" = '0' ]; then
  exec gosu postgres "$BASH_SOURCE" "$@"
fi


if [ -z "$DATABASE_ALREADY_EXISTS" ]; then
  echo " db is missing"
  docker_verify_minimum_env
  docker_init_database_dir
  pg_setup_hba_conf

  # only required for '--auth[-local]=md5' on POSTGRES_INITDB_ARGS
  export PGPASSWORD="${PGPASSWORD:-$POSTGRES_PASSWORD}"

  docker_temp_server_start "$@" -c max_locks_per_transaction=256
  docker_setup_db
  docker_process_init_files /docker-entrypoint-initdb.d/*
  docker_temp_server_stop
else
  echo " db already exists"
  docker_temp_server_start "$@"
  docker_process_init_files /always-initdb.d/*
  docker_temp_server_stop
fi

echo " .. starting!"
exec postgres "$@"

This is how I use postgres on my projects and preload the database.

file: docker-compose.yml

  db:
    container_name: db_service
    build:
      context: .
      dockerfile: ./Dockerfile.postgres
    ports:
      - "5432:5432"
    volumes:
      - /var/lib/postgresql/data/

This Dockerfile load the file named pg_dump.backup(binary dump) or psql_dump.sql(plain text dump) if exist on root folder of the project.

file: Dockerfile.postgres

FROM postgres:9.6-alpine

ENV POSTGRES_DB DatabaseName

COPY pg_dump.backup .
COPY pg_dump.sql .

RUN [[ -e "pg_dump.backup" ]] && pg_restore pg_dump.backup > pg_dump.sql

# Preload database on init
RUN [[ -e "pg_dump.sql" ]] && cp pg_dump.sql /docker-entrypoint-initdb.d/

In case of need retry the loading of the dump, you can remove the current database with the command:

docker-compose rm db

Then you can run docker-compose up to retry load the database.