Do we consider our relatives' visits to us as fulfilling the ties of kinship, or is initiating the visit from our side a condition for that, especially with obstacles preventing visits to some relatives?

If your relatives visit you, they are the ones who are maintaining ties, and the ties are not truly maintained by you unless you visit them or contact them. Maintaining kinship ties (silat al-rahim) is determined by custom, and it does not have a specific method or time prescribed in Islamic law. There are varying degrees of maintaining ties, the lowest of which is abandoning estrangement and greeting one another. You are obligated to do what you can to maintain ties according to custom, and what you are unable to do, you are not held accountable for.