It’s a quite biblical theme: redemptive suffering, or joining our suffering with that of Christ (see several papers of mine on it, below); yet Protestants often do not address this scriptural theme, and claim that such a view denies the sole salvific / redemptive sufficiency of the cross and sacrifice of our Lord Jesus.
In any event, those who ignore this biblical motif and practice have to alternately explain several Bible passages that explicitly teach it. The Anglican C. S. Lewis (1898-1963): considered the most popular Christian apologist of the 20th century, did accept this notion.
From: The Collected Letters of C. S. Lewis, Vol. III: Narnia, Cambridge, and Joy 1950-1963, edited by Walter Hooper, HarperSanFrancisco, 2007:
It is very remarkable . . . that you should write about our vicarious sufferings . . .
I have not a word to say against the doctrine that Our Lord suffers in all the sufferings of His people (see Acts 9:6)
[actually, he refers to Acts 9:4-5 (RSV) And he fell to the ground and heard a voice saying to him, “Saul, Saul, why do you persecute me?”  And he said, “Who are you, Lord?” And he said, “I am Jesus, whom you are persecuting;”]
or that when we willingly accept what we suffer for others and offer it to God on their behalf, then it may be united with His sufferings and, in Him, may help to their redemption or even that of others whom we do not dream of. . . . The key text for this view is Colossians 1:24.
[Colossians 1:24 Now I rejoice in my sufferings for your sake, and in my flesh I complete what is lacking in Christ’s afflictions for the sake of his body, that is, the church,]
Is it not, after all, one more application of the truth that we are all ‘members of one another’?
[Romans 12:5 so we, though many, are one body in Christ, and individually members one of another. ]
I wish I had known more when I wrote the Problem of Pain. (To Mary Van Deusen, 12 September 1951, pp. 134-135)
[S]uffering can (but oh!, with what difficulty) be offered to God as our part in the whole redemptive suffering of the world beginning with Christ’s own suffering. . . . sufferings . . . can be so taken that they are as saving and purifying as the voluntary sufferings of martyrs & ascetics. (To Mrs D. Jessup, 5 January 1954, p. 405)
Of course we have all been taught what to do with suffering — offer it in Christ to God as our little, little share of Christ’s sufferings — but it is so hard to do. . . . I suppose the great saints really want to share the divine sufferings and that is how they can actually desire pain. But this is far beyond me. (To Mary Willis-Shelburne, 26 April 1956, p. 743)
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Photo credit: Stigmatization of St. Francis (bet. 1295 and 1300), by Giotto (d. 1337) [public domain / Wikimedia Commons]