The reason checking in fails is that it asks the recipient to do the work. They have to remember the context, decide whether anything changed, and compose a reply, all in service of your process rather than their situation. A message containing something useful gives them a reason to open it and something to respond to.
Useful is broader than it sounds and does not require producing anything new. An answer to a question they raised. Something that changed in your industry affecting them. A relevant example from work you did for somebody comparable. Or a straightforward observation about their situation that occurred to you afterward. Each of those is a message they would have been glad to receive from anybody.
Trigger on behaviour rather than on elapsed time wherever you can. Somebody who requested a quote, visited a pricing page, or asked a specific question has told you something, and a follow up referencing that reads as a continuation. A message sent because a week passed is about your calendar rather than their situation, and it reads that way.
Give them a route to slow it down that is not silence. Asking whether the timing is wrong, and when to come back, converts somebody who would have disengaged into somebody who tells you March. That information is worth more than the follow up itself, and almost nobody asks for it because it feels like inviting a delay.
Decide the number of messages in advance and stop when you reach it. The behaviour that generates genuine irritation is indefinite pursuit, and it is usually the result of a sequence with no exit. Three or four messages over a few weeks establishes whether there is anything there, and continuing past that produces the impression you were trying to avoid.
Vary the channel rather than the frequency. Three emails to somebody not reading email produces nothing except a record of you chasing. One phone call, or a message on whatever they actually use, resolves in minutes what a fortnight of emails did not.
Watch what happens after the sequence ends, because the correct destination is not deletion. Somebody who was not ready is frequently ready later, and moving them onto whatever ordinary material you send keeps you present without pursuing them. That is the difference between a lapsed prospect and a lost one.
Then judge it by replies rather than by open rates. A sequence with respectable metrics and no conversations is one people are tolerating rather than reading, and the response rate is the only measure that tells you whether the messages were actually welcome.
Space the messages further apart than feels natural, because the anxiety about being forgotten belongs to you rather than to them. A fortnight between messages reads as attentive while three days reads as pursuit, and the person deciding is usually working on a considerably longer timescale than the person waiting.
Write the sequence before the first message rather than composing each one as the silence extends. Deciding in advance what each message contains prevents the pattern where the first is useful, the second is a reminder, and the third is a request, which is the shape that irritates people.
Exclude anybody who has already replied, which sounds obvious and is the most common failure in small business sequences. Somebody now in a conversation with you should not continue receiving the automated messages, and receiving them signals clearly that nobody is paying attention.