Identify user plan behavior switches for subscription transaction in dplyr

I have a df identifying users and their recurring transactions in a subscription-based business model, where transactions can happen on a monthly, 3 months or 6 months cycle (every 30, 90, 180 days respectively).

library(tidyverse)

df <- tibble::tribble(
        ~user_id,        ~date,      ~plan,   ~order_type,
               1, "12/23/2018",  "monthly", "acquisition",
               1,  "1/22/2019",  "monthly",      "repeat",
               1,  "2/21/2019", "3-months",      "repeat",
               1,  "5/22/2019", "3-months",      "repeat",
               1,  "8/20/2019", "3-months",      "repeat",
               2,  "5/18/2019",  "monthly", "acquisition",
               2,  "6/17/2019",  "monthly",      "repeat",
               2,  "7/17/2019",  "monthly",      "repeat",
               3,  "9/14/2018",  "monthly", "acquisition",
               3, "10/14/2018", "6-months",      "repeat",
               3,  "4/12/2019", "6-months",      "repeat",
               4,  "9/14/2018", "3-months", "acquisition",
               4, "12/13/2018",  "monthly",      "repeat",
               4,  "1/12/2019",  "monthly",      "repeat"
        )

Each customer (identified with a unique user_id) is free to start with any of those "plan" and over the course of her/his subscription, can move from one plan to the other.

My objective is to identify - with a preference on Dplyr solutions:

• User upgrades (namely, her/his first order (= acquisition) is "monthly" but the subsequent are either "3-months" or "6-months")
• User downgrades (namely, her/his first order (=acquisition) is either "3-months" or "6-months" but the subsequent are "monthly")

My desired output is:

df_output <- tibble::tribble(
               ~user_id,        ~date,      ~plan,   ~order_type, ~behavior_type,
                      1, "12/23/2018",  "monthly", "acquisition",            "-",
                      1,  "1/22/2019",  "monthly",      "repeat",            "-",
                      1,  "2/21/2019", "3-months",      "repeat",      "upgrade",
                      1,  "5/22/2019", "3-months",      "repeat",            "-",
                      1,  "8/20/2019", "3-months",      "repeat",            "-",
                      2,  "5/18/2019",  "monthly", "acquisition",            "-",
                      2,  "6/17/2019",  "monthly",      "repeat",            "-",
                      2,  "7/17/2019",  "monthly",      "repeat",            "-",
                      3,  "9/14/2018",  "monthly", "acquisition",            "-",
                      3, "10/14/2018", "6-months",      "repeat",      "upgrade",
                      3,  "4/12/2019", "6-months",      "repeat",            "-",
                      4,  "9/14/2018", "3-months", "acquisition",            "-",
                      4, "12/13/2018",  "monthly",      "repeat",    "downgrade",
                      4,  "1/12/2019",  "monthly",      "repeat",            "-"
               )