You may want to check the ALB defaults and adjust these, also check the deployment type settings, which you may want to adjust depending on what the desired count is for the service. I believe the minimum healthy percent is 50% and maximum 200%, by default.
For ALB check the health check settings, but perhaps in particular the deregistration delay, which is 300 seconds by default. Also, do you have any delay in the task definition for start and stop delays of the containers?
Some of these settings are probably more suited to various software running on regular EC2:s than containers, so it might be worthwhile to tweak them.