Two ways to follow up
Message Sender sends one message to the people in your lead list who have accepted, when you launch it. You choose the day. Outreach sequences do it on their own: a connection step, a wait step of however many days you set, then a message step, for every lead you enrol.
Both check that the person is a first-degree connection right before sending. Anyone who hasn't accepted is skipped and waits for a later run or a later check.
How the wait step works
The wait counts from the connection step, not from the acceptance. If you set three days and the person accepts on day one, the message goes out on day three. If they accept on day five, the message step finds them connected at its next check and sends then. Message steps are checked every 12 hours, so a message goes out within half a day of the person becoming a connection once the wait is over.
Messages share the daily cap of 30 with Message Sender. A step that would go over the cap waits and is tried again about 12 hours later.
How long to wait
A few days. The day someone accepts, they have just seen your name and your note; a message that lands within the hour reads as a script firing. A message three to five days later reads as a person getting round to it. Longer than a week or two and they have forgotten who you are, so the message has to re-introduce you, which makes it longer and worse.
If you reach people who are busy in a particular way, think about their week rather than a rule. There is no number that works for everyone, and we don't pretend to have one.
How many follow-ups
One. Maybe two, far apart. The sequence checks the conversation before each message and skips anyone who has replied, but that check has not been proven against live LinkedIn yet, so don't lean on it: a long sequence is a long list of chances to write to someone who already answered. Keep it short and read your inbox daily.
What to write
Thanks for connecting, then one question. Not a deck, not a link to book a call, not a paragraph about your company. Templates can use {name}, {company} and {headline}, up to 1,000 characters, and short is still better.
Thanks for connecting, {name}. Quick one: how does {company} handle month-end reporting today, spreadsheets or something else?
A question gives the person something easy to answer. An offer gives them something to decline.
If they never accept
Their message step keeps waiting and is checked every 12 hours. They never get the message unless they accept, which is as it should be. You can delete them from the lead list whenever you like.
Using Message Sender by hand
Message Sender has no wait step; you are the timer. A simple habit is to run it once a week for the people who accepted since the last run. It checks each person is a first-degree connection right before sending, waits a random 20 to 60 seconds between messages, and stops at the daily cap. It can message the same person again on a later run, so use it once for each batch of new connections rather than on the whole list every time.
The arithmetic takes care of itself: at most 20 requests go out a day, and the message cap is 30, so follow-ups never fall behind the people who accept.
After the reply
Answer yourself, in LinkedIn. Nothing here writes replies, and nothing should: the follow-up's job is to start a conversation, and conversations are yours.