Your family members who are not EU citizens cannot be needed to apply for a residence card verifying their right to live there in their first 3 months in your host country. While in various countries they may have to give an account of their presence upon arrival.
Your non-EU family members must enlist their abode with the appropriate authorities (often the town hall or local police station) after 3 months in your host country.
To get hold of a residence card, they will require:
A legitimate passport.
Your registration certificate as an EU citizen or any other evidence of your dwelling in the country.
Evidence of the family relationship with you (like a birth or marriage certificate).
For (grand) children, confirmation that they are under 21 or dependent on you.
For (grand) parents, verification that they are dependent on you.
For other family members, evidence that they are dependent on you or there is severe health ground needing you to take special care of them.
For unmarried buddies, evidence of a long-term or stable bond with you.
No other credentials may be asked for.
The system should make their judgment to hand out a residence card or not in 6 months. In any case, your non-EU family members cannot be disqualified if their visa ended while their request is being processed.
If their request or application is declined, the government department must provide them the judgment in writing, declaring the grounds for the judgment and its inferences. The authorities must also state how your relatives/partners can request again and at what time.
If their application is approved, the residence card will often be handed out for free. If the amount is charged, they may not be more than those costs to nationals for the same credentials, like identity cards.
The papers should declare that it is the residence card of an EU citizen family member.
The residence card should be applicable for 5 years (or for your considered duration of stay, if it is for a brief time). If there is any change of address, its details must be reported to the authorities.
The residence cards may be used by your family members to travel to another EU country. But if they want to shift to another EU state, they have to apply for a new residence card in the new host country. And provided they are dependent on you, you have to shift with them.